I've been seeing discussions about how great weapon fighting and even dueling, when reaching a certain point, are stronger than Two-Weapon Fighting. What are the advantages & disadvantages of the Two-Weapon Fighting option as compared to the other options?
The defense fighting style gives +1 AC while wearing armor. The only advantage this has is that you can get it earlier than your first feat, but only by a few levels. At level 4, you can take the dual wielder feat, and then two weapon fighting will have evened out the AC difference. Your main disadvantage at this point with TFW is not having a shield, but you will do more damage from here on out as a trade off.
You will have +2 AC because of the shield you need to use the protection effect. However it requires you to stand very close to an ally and doesn't actually do any damage. Plus, it's entirely situationally dependent. While disadvantage on an opponent's attack sounds good, you can actually save your allies from taking damage by ending the fights earlier, which is done with the extra damage from two weapon fighting. Protection only makes fights last longer, so your friends have more time to be hit with more damaging abilities. In most respects, this means two weapon fighting is the better choice here.
Vs. Great Weapon Fighting
As described in this answer, Two weapon fighting has the advantage in damage at the early levels. I won't copy over all of the math, but the crux of things is that re-rolling 1s and 2s tends to add about 1 damage on average to your attacks. Compare this to adding your whole ability modifier to damage, and at the early levels, you can expect the two weapon fighter to deal about 1 extra damage per round. However, as the fighter gains extra attacks, the benefits from GWF (and the almost mandatory feat Great Weapon Master) apply to each attack that the fighter can make. The Great Weapon Fighter will end up dealing About 10 more damage per round because of the benefits being applied to each of the 4+ attacks the fighter can make. So, as a GWF fighter gets extra attacks, TWF starts falling farther behind in damage. Though, it's worth noting that you will have +1 AC with the dual wielder feat.
TWF has the advantage when it comes to damage. Though you will ultimately have -2 AC compared to a dueling fighter because of their shield (-1 with dual wielder). The damage bonus from dueling applies to each extra attack that the fighter will get, just as with GWF. So, duelers end up with a sizable damage boost when they have extra attacks, which minimizes the difference in damage between the fighting styles. Additionally, a dueler gains a lot of versatility with the shield master feat. Knocking an enemy prone with a bonus action can give you advantage on attacks, and you have extra survivability on dexterity saves, which are one of the most common saves in the game. Using the formula and assumptions from the link on GWF damage, we can average the damage of a longsword dueling champion fighter at 20th level as 4×(11.5×0.8+.15×4.5) = 39.5. A TWF dual wielder with rapiers will deal 5×(9.5×0.8+1.5×4.5) = 41.375. So at the cost of about 2 damage per round, A dueler will get +1 AC, and the shield master feat.
The two are used for completely different play styles. Obviously, if you plan to use ranged equipment primarily, then archery is better. If you plan to mostly be up close, TWF is better. Each fighting style would pretty much be useless outside of niche cases when used with the opposite playing style.
When it comes to the other fighting styles TWF deals the most damage in melee range at the early levels. As a fighter starts getting extra attacks though, in pure damage TWF is outclassed by GWF because of the boosts to each attack. Dueling outclasses TWF for all around generic usefulness because of the fairly even boosts to offense and defense, as well as the utility provided by shield master. So, while you have a damage boost with TWF in the early levels which may come in handy for a one-shot or short-lived campaign, dueling or GWF tend to be better choices if you expect your character to have a very long career. Excluding any character flavor, of course.
Break down, the benefit of Dueling is having no weapon in the other hand. For a Paladin or melee spell caster who needs his hand free to cast, that means you gain +2 damage. Also you can use a shield to gain +2 AC. So basically it's a +2 damage, +2 AC bonus.
Great Weapon Fighting is great when used with a maul or greatsword due to the fact they're both 2d6+modifier damage, meaning — at minimum — you're doing 2+modifier damage. But you can re-roll that minimum roll, so chances are you will do more damage with Great Weapon Fighting. But there is the drawback of the −2 AC compared to Dueling with a shield.
Also you're allowed to reroll ones or twos on crits with Great Weapon Fighting, so that leads to a bigger increase.
The TWF style is also very good if you want to dual-wield hand crossbows. You will need the feat called Crossbow Expert to pull this off, but a human fighter will be able to do it at first level. Get the sharpshooter feat at level 4, and you're good to go. Maybe put 2 levels in Ranger or Blood hunter (if your GM allows that class) to get the Archery combat style for a cheesy way to off-set the -5 attack penalty from sharpshooter. At level 5 you will be popping off three crossbow shots per round from 120 feet away and getting at least +13 damage. Pretty sweet if you ask me.
If you prefer melee combat, though, a two-handed weapon and the Great Weapon Master feat will be much better. The GWF style would be your best option for a 2-handed warrior, but the Protection style is a good choice, too, especially since you won't be using a shield