(This question is a comparison to 3.x, though things might have been different in 4e)
In 3.5e there is a large power and capability gap between fighters and wizards that fighters couldn't hope to close, even in their nominal area of excellence. Is this problem still around?
Short Answer:
Basic Evocation Wizards are now the equivalent of Tier 3 at lower levels.
Basic Champion Fighters are now the equivalent of Tier 3 at lower levels.
The Usefulness of Tiers will have to change for 5e, as tiers 1, 5, and 6 at this time don't exist, and likely won't exist as the overall power of 5e is much flatter. We can compare classes by looking at their average DPAD (damage per adventuring day).
For overall power however, it's going to really depend on level.
Thanks to the work by @Waxeagle we can see how power via damage shifts over level.
Basically, at level 11, the Fighter is doing more damage than the evocation wizard who is focused solely on damage. However, by level 17, both the cleric and the Wizard have enough spell slots to outperform the Fighter and Rogue. In addition, level 9 spells are by their nature "game changers." It should be noted that at level 20, with the third extra attack, the fighter shoots up in power as well, though not enough to outperform spells such as meteor swarm, the nuclear weapon of D&D. One caveat to the current graph is that not all levels are shown; only the level when the "adventure tier" changes. For example, not shown on the graph is level 10, where the fighter still only has one extra attack, the wizard and cleric are doing about the same amount as the fighter.
So while it is still the case at the highest levels that magic is more powerful than the mundane, up until the epic tier, this problem does not really present itself.
This obviously doesn't take into account the non-damaging effects of spells, such as haste, or invisibility, or flying, etc. However, with the new concentration mechanic, these spells are situational and cannot be easily piled on top of each other as they could in 3rd edition, to create a quadratic power curve.
Relevant data for the chart:
The above chart is based on the various classes and an unlimited number of enemies with the following defenses; No assumption is made about the HP of the monster, so damage is "unbounded" in the terms of the spreadsheet:
Dex Save 2
AC 15
Con Save 2
However, If I change the defences of the monsters to something less reasonable in the game, such as giving a Dex and Con save of 5 and an AC of 10, we see the fighter keeping up with the wizard and again surpassing it at level 20.
You can compare these 5e graphs, to this 3.5 graph and notice how the fighter is at the bottom of the graph and stays very linear, while the wizard has a slight a quadradic curve in power gains.
(Graph taken from here)
Tier 1: Capable of doing absolutely everything, often better than classes that specialize in that thing. Often capable of solving encounters with a single mechanical ability and little thought from the player. Has world changing powers at high levels. These guys, if played well, can break a campaign and can be very hard to challenge without extreme DM fiat, especially if Tier 3s and below are in the party.
It's pretty clear that currently, the Evocation Wizard can not do absolutely everything. Because of the concentration mechanic, the Wizard can really only do one cool thing at a time. You can't fly and be invisible at the same time without another magic item or wizard casting those spells on you, for example. Secondly, while some spells can duplicate other class abilities, such as "knock", those spells are only situationally better, and come at a high resource spot (limited spell slots) that other classes lack. So the Wizard is no longer a tier 1 class.
Instead, I would say that Tier 3 fits the Evocation Wizard.
Tier 3: Capable of doing one thing quite well, while still being useful when that one thing is inappropriate, or capable of doing all things, but not as well as classes that specialize in that area. Occasionally has a mechanical ability that can solve an encounter, but this is relatively rare and easy to deal with. Challenging such a character takes some thought from the DM, but isn't too difficult. Will outshine any Tier 5s in the party much of the time.
The bolded part is why I believe a Wizard is Tier 3. A wizard can, with the right spells, do anything another class can do (with some exceptions), but it can't do those things as well as the class that specializes in that thing. An invisible wizard is hidden but can't sneak. A wizard with damage spells can't kill a single high hitpoint target as well as a fighter can. A Rogue will have better skill abilities when it specializes in those skills than a wizard can ever get.
Tier 5 is defined by:
Tier 5: Capable of doing only one thing, and not necessarily all that well, or so unfocused that they have trouble mastering anything, and in many types of encounters the character cannot contribute. In some cases, can do one thing very well, but that one thing is very often not needed. Has trouble shining in any encounter unless the rest of the party is weak in that situation and the encounter matches their strengths. DMs may have to work to avoid the player feeling that their character is worthless unless the entire party is Tier 4 and below. Characters in this tier will often feel like one trick ponies if they do well, or just feel like they have no tricks at all if they build the class poorly.
No class is good at only one thing. Because of the Background system, all classes and characters are able to focus on skills that are outside of their class. So while you can make a fighter that is only good at fighting and can't do anything during exploration or social interactions, it's more likely to build a fighter that is good at fighting and leading people. Or good at fighting and using certain tools. Because of Bounded Accuracy, all characters are decent at whatever skill they are proficient in. The champion fighter is best built for two-handed heavy weapons, but they can also be a good archer or duelist. Fighters excel in combat and, at higher levels, can move across the battle field, hitting multiple creatures in one turn in the same way that a wizard might target more than one creature with a spell. They are also able to survive situations that others would not without a cleric. So I don't think they can be called Tier 5.
Champion fighters are however tier 3, because they are
Capable of doing one thing quite well, while still being useful when that one thing is inappropriate.
Fighters do combat really well, they survive well and can kill well. However sometimes a situation will be called where there is no combat. Fighters can excel at jumping across chasms, or climbing/swimming out of tough spots more so than wizards and clerics. With Background again, they can also call upon their traits. In the starter set for example, the Fighter is a folk hero, who is able to rally others to their cause.
However, the tier system as a whole is a bad way to judge D&D 5e classes. The power level of the entire system has been made "more flat", and there will need to be better definitions of how to compare classes.
The progression of Wizards was smoothed out quite a lot, and Fighters were upgraded.
They became more flexible, but weaker in damage at the same time.
Tier 4 on levels 1-4
Tier 3 on levels 5-10
Tier 2 above
In 3.5 and earlier if you memorized Magic Missile and Burning Hands, you could cast both once per day. Now you can also cast one of them twice, and the other not at all. For a cast that was too versatile to begin with, I would not call this progress. Because of this versatility however there is no point of talking about an Evocation wizard any more. At most half of your spells need to be damaging, so you can still fly and teleport and solve any other problem.
On the other hand, earlier if you were level 10 instead of 5, you not only got more Fireballs, but they were stronger as well. This made Wizards Quadratic. Now this is handled, quite well actually. Your level does not matter when you cast a damaging spell, but you can cast it in a higher level slot for more damage. So either your Fireballs are more numerous or stronger. I like this one.
Champion Fighters are as boring as ever, one attack identical to the next, Battlemaster Fighters have much more options, making them quite versatile in combat.
Bow Fighter is mostly viable now. Still the two-weapon fighter is the most damaging option on levels 1-4, later they become the weakest option.
Tier 4 on level 1-4
Tier 3 above
Using just at-will attacks, a Wizard is significantly behind a Fighter, doing only about half as much damage. The problem is that Fighters do nothing but at-will attacks while Wizards get more and more spell slots as they level up, so at the end they hardly need to use at-wills.
The Fighter just got 3 attacks, for a great weapon user it is about 3 x (8.33+5) = 40 damage if all hit. Assuming a 50% hit chance, it is 20 damage per round. With Great Weapon Master he could go 50% higher, but only with Advantage, and he needs outside help for that.
The Wizard Firebolt does 16.5 now, quite far behind.
However, he can cast Sunbeam now, doing 27 damage on a failed save, or half otherwise. Assuming a 50% save chance, it 20.25 damage. If there is more than one enemy, he can deal even more in total, until the end of the encounter.
Now he is out of 6th level spells, but a Fireball does just as much damage, and he has slots for 8 of those. Seems more than enough for the rest of the adventuring day.
Fighters are really good against single opponents, but Wizards slowly reach them in this regard on high levels, while being wildly superior against groups, and much more useful outside combat.
Yes, fundamentally the problem still exists is 5th edition Dungeons & Dragons, but with some changes.
At low levels, especially level 1, the wizard does not have the previous disadvantage of being able to cast just one or two spells and then being useless for the day. He now has "cantrips" that work like 4E at-will spells and deal up to 1d10 damage, usable every round. His level 1 spells (two per day at level 1 plus an option to regain by a short rest) are significantly more powerful than in previous editions, for example burning hands now deals 3d6 to multiple enemies, half on saving throw. Magic missile has no attack roll nor saving throw and deals 3d4+3, and you can split those three missiles among multiple enemies. The level 1 fighter in comparison, taking the starter set characters as example, has an attack bonus about the same as the wizard for spells that use attack bonuses, and deals 1d12+3 damage. So in a typical level 1 combat where the wizard casts one level 1 spell plus cantrips, and the fighter uses his weapon attack all the time, the overall damage output is going to be about equal.
From that base the "quadratic wizard" issue still happens: With every level the wizard gets more spells, and every two levels he gets more powerful spells in addition to that. Also he can cast his lower level spells using higher level spell slots for greater effect. He can potentially deal a lot more overall damage than the fighter if you consider that for example the 8d6 fireball isn't just hitting one monster, while the fighter is limited to single-target attacks in the basic rules. More importantly, all those spells give the wizard tons of options, also in out-of-combat situations with spells like invisibility or fly.
The fighter is a bit less "linear" than he was in previous editions. He gets more bonuses by leveling up, gaining extra actions or attacks, faster ability score improvements (which can be spent on feats, as yet not described), martial archetype features, and saving throw rerolls. What he doesn't gain, at least not in the basic rules, is a more options. He can hit multiple times, and really hard, but other than attacking with his weapon or dodging his options are very limited. Potentially the new feats will improve that situation, but they aren't in the basic rules yet.
(I answered here earlier, but I do not agree with that answer any more)
On first level a Fighter does around 7.5-13 damage per round (Protection and Two-Weapon Fighting styles, respectively). A Wizard does 5.5 with Fire Bolt, or 6.5 with a Light Crossbow (cantrip are this bad)
Level 20 is a big one for Fighters, not so much for Wizards. So to make a point, I pick level 17 for comparison. This is where spellcasters get their most powerful spells, and they will never have more than one.
A Fighter can do 3 times 2d6+5+10 = 70 damage with the Great Weapon Master feat, using the Great Weapon Fighting Style. With Action Surge it becomes more than Power Word Kill, and equals Meteor Shower. The later can damage several foes, but the Wizard has to wait a long rest to repeat it, while it takes just a short rest for the Fighter. With the Riposte maneuver, that the Fighter can use 6 time between short rests, it goes even higher, not to mention the bonus action attack from GWM, that is bound to happen sooner or later with this much damage.
Wizards do get much more flexible with levels, and generally more powerful in control, but mostly not in raw damage. Evard's Black Tentacles provides great control for example, but negligible damage.
However, as the number of slots and number of spells prepared go up quickly, the Wizard's own effectiveness does not follow. Between Legendary Saves, Magic Resistance and resistance to the most common damage types, it is often not even worth trying to affect the enemy.
Quite frequently the most useful thing they can do is to provide support to the Fighter with Haste, Improved Invisibility or Fly.
Paladins are like Fighters, while Bards, Clerics and and Land Druids are similar to Wizards. The rest is somewhere in between.
If you build for power to eliminate an enemy, especially a single boss, Wizard is not your best bet any more. They deal spectacularly with lots of smaller ones, but are no longer capable to outshine Fighters in every situation.
Quite the opposite.
Yes, it is still a problem.
A lot of the answers here miss what the problem is, though.
Wizards are controllers - that is to say, their primary job is controlling the battlefield. Indeed, all of the best casters are controllers.
A lot of people think that the real strength of a wizard is spells like fireball. And, yes, fireball is hideously powerful - it does huge damage to a big AoE, thus wildly outclassing the fighter in terms of damage against multiple opponents.
However, wizards have even stronger spells. Spells that totally disable an opponent - that take away their actions entirely - effectively give every other combatant an extra round of actions. If you're fighting a boss, and stun them for a round, that basically means you gave everyone else in the entire party an extra action - which by definition is better than what the fighter is doing, because the fighter's own action is going to be included in that! Spells that disable entire groups of enemies are even more potent. And other spells can do similarly broken things - splitting an enemy group in half so you first fight one group, then another, is turning one hard encounter into two easy ones.
Moreover, wizards can often bypass encounters entirely with their magic, which is obviously better than anything else, as they basically "beat" everything in the encounter by themselves - sometimes with as little as a single spell.
But on top of that, wizards also have enormous out of combat vesatility as well - their spells aren't only useful in combat, and indeed, it is outside of combat that they really end up most overshadowing non-casters.
The same is true of other casters, for that matter - indeed, this is part of why the bard is so powerful: they get a lot of these controller and versatility effects. And clerics can fight in melee almost as well as fighters, on top of being very powerful casters.
That's not to say it is as bad as it used to be - casters are a lot weaker in 5th edition than they were in 3.x. But they still overshadow the non-casters, especially fighters. Oftentimes, the non-casters are really only capable of doing one thing well, while the casters are capable of doing lots of things well, and this is honestly a much larger problem than the fact that wizards are better than fighters in combat.
Hitpoint damage is the weakest interaction and least important metric of character ability, which is why fighters, who have no other means by which to meaningfully interact with the game's systems, will always be the least useful class at all levels.
The only power that matters is narrative control: the ability to solve problems, to define the group's strategy, and to direct the overall direction of the story.
As long as casters can fly, create planes of existence, bind extradimensional beings to their service, discover the answer to any question, scry on anything anywhere, and to alter reality to their whim - and also have the only meaningful defenses against their own infinite versatility - other classes will to continue to not matter.
