The rules say that creatures in a heavily obscured area "effectively suffer from the blinded condition".
The blinded condition states:
Attacks rolls against the creature have advantage, and the creature's attack rolls have disadvantage.
So when the Darkness spell (to use one example) is cast, combatants gain advantage and disadvantage when attacking others who are also inside the area, which cancel out.
Since they then cancel each other out and attacks are made normally, what really changes?
Casting darkness will cancel out both advantage and disadvantage, due to the way they stack.
If your opponent has advantage and you have disadvantage, then cancelling both will be good for you, and bad for them.
Some characters (eg, Warlocks with Devil's Sight) can see normally in magical darkness.
So, their opponent will be blinded, but they won't.
For example, you cannot:
Use Protection fighting style.
Use Uncanny Dodge.
Make Opportunity Attacks.
Cast any spells that require a target you can see.
Your attack rolls will be at normal (the advantage for the target not seeing you is negated by the disadvantage for not seeing your target) but the combat will play differently because of these unusable features. In particular, fights in the dark will usually be more mobile since you don't have to worry about opportunity attacks.
This is assuming everyone is in the dark! If one combatant can see in the dark then the combat will be very different (and probably very deadly).
You're right that if both combatants are just stumbling around blind, there's no net benefit, but if one of the combatants can see in the dark, using Darkvision or Devil's Sight, then they suddenly gain a significant edge in combat, because they get advantage in attacks against their enemies but their enemies have disadvantage against them.
In terms of the fiction, you can imagine that there are two blinded people fighting: they don't know where to strike, but they also don't know where the strikes are coming from. Thus, the relative advantages and disadvantages cancel out, and you're back to a neutral contest. However, if one of the combatants can see and the other can't, the one that can see is going to have a much easier time in attacking and defending.
Effectively, darkness imposes a circumstance in which seeing in the dark actually confers some kind of bonus, as opposed to fighting in bright light, where that ability does very little.
Being unseen allows creatures to hide. Targeting a hidden creature requires you to guess where they are. A rogue who can use a bonus action for cunning action to Hide and Attack every round is going to have an edge over a creature that must choose to Hide or Attack.
To clarify, an unseen creature is not hidden - everyone knows exactly where the creature is, they do not have to guess. How they know is irrelevant, maybe they can hear, smell, or taste them: whatever.
An unseen creature has met the preconditions to take the Hide (bonus) action. If they do so they are hidden from every creature whose passive Wisdom (Perception) score is less than their Dexterity (Stealth) check. Remember that for a creature that relies predominately on sight (i.e. most creatures) their Wisdom (Perception) check is at disadvantage, for a passive check this is a -5 penalty. That is, whatever senses they used to know where the unseen creature was are inadequate to locate the hidden creature. Of course, if they rely exclusively on sight (I don't think there are any) then they automatically fail the check as per the Blinded condition.
If you move then these creatures do not know where you are - they have to guess if they want to attack you - only if they guess correctly are you at risk but they still have disadvantage on the check (and advantage if you can't see them). Alternatively they can spend an action to Search and engage their active Wisdom (Perception) to try and find you but this probably means giving up an opportunity to do damage this turn: if you can hide again on your turn, they have effectively done nothing.
In addition to what has been said above, creatures that do not rely on sight, are immune to the blinded condition or can see in magical darkness will be able to attack without any disadvantage. Thus allowing them great advantage over those that do not.
a warlock with Devil's sight can use darkness to make enemies have to guess where it is while it has free reign to do as it pleases. A villain in charge of a dungeon could cast darkness on the party allowing the slimes to do their work, etc. and in a place filled with traps, the denizens would know where the traps are while the party can't see them and are far more likely to fall into them.
creatures that have sunlight sensitivity can also use magical darkness to level the playing field during the day.
In addition to all the other answers, which cover the big details, there is another key difference.
Advantage and disadvantage to attacks do not always even each other out, there can be situations where the advantage/disadvantage tradeoff is useful even without an ability to see through darkness.
The big benefit comes from placing your target in darkness but not yourself. Your target is blinded no matter what, You are effectively 'blinded' if you attack your enemy in the darkness as well, but your enemy has disadvantage when attacking you through the darkness and if you attack him with a long range attack your disadvantage even's out with his advantage to give you a net gain.
Thus any situation where your enemy has long range attacks you can benefit from casting darkness on where they stand. In the best case scenario, where they can't move out of the darkness, you now have forced disadvantage on every range attack they do for the entire fight. Worse case scenario you force everyone in the darkness to spend an action to move out of the darkness wasting actions.
So some situations in which it's useful to put your enemy in darkness:
Imagine fighting in a courtyard with enemy archers attacking form some overhead wall. Casting darkness on them would work wonders, it wouldn't affect you but they would either have to spend several rounds moving down to the courtyard, and closer to you, or put up with constant disadvantage.
You won't make any attack rolls, you just want to survive the volley of arrows/spells until you make it to safety
Maybe one or more of your party members just sustained some major injury and is near death. You're enemy goes next and is going to try to finish them off. If you have no other option a darkness spell can increase the odds that your team survives the next attack so it can heal/regroup.
Alternatively maybe the enemy is attacking civilians or your on an escort quest or simply is smart enough to all focus their ranged attacks at one vulnerable ally who's doing something plot critical. The situation is all the same, at the moment you want those people safe.
Any situation in which a total defense action would make sense, darkness would as well. Since total defense would not be an attack roll, it means you get high defense and your enemy disadvantage giving you a very good chance of negating every attack.
Imagine a situation where your fighting or doing something else important in a particular room, and know that a large number of slightly lower reinforcements will be summoned when you do that could easily overwhelm your team before your done. You have your Warrior stand in front of the door, making total defense actions, and cast darkness so it covers just the doorway. Now instead of fighting a large force that could easily defeat you your highest defense teammate with a total defense boost and advantage on defense has to guard against the attack of only one foe that can fit in the doorway, and the enemy may already be just low enough to have a lower attack. Your warriors defense is already higher then enemies attack and disadvantage favors whoever has the statistical advantage so your warrior can pretty much stall those reinforcements indefinitely buying you a long time to do whatever you need to do.
Your enemy is about to read the ancient runes of evil to summon the demon from beyond and you just need a few more rounds to stop him, cast darkness on those runes and you have that time. Lots of skill checks have to happen at a specific place and blinding that area keeps your enemy from doing whatever they intended to do there.
There are plenty of uses for darkness that aren't explicitly spelled out in RAW but which any GM is likely to allow. For example it increases stealth and ability to sneak around, though another answer already mentioned that.
You could use it to spook or scare someone. Or to obscure your face when negotiating with someone whom you don't wish to know your identity, or to prevent someone from making some sort of check that requires eyesight that it's imperative he fails to make at this moment, etc. There are a million role-playing uses for various situations.
