Both the Paladin and the Warpriest are combat-oriented divine casters, but what I want to know is, from an optimizer's perspective, what is the difference between the two classes in practice? What ways do the classes deal with challenges in-combat, and how do they deal with out-of-combat encounters? In asking this question, I am trying to determine the differences and merits associated with two classes that appear to be trying the fill the same niche in a way.
[RPG] Differences Between Warpriest and Paladin
It really depends. for a cleric, they are functionally identical.
Here are the damage expressions:
Two hander: 2d6 + str
TWF: 1d6 + str, 1d6
Literally exactly the same when you factor in that your to-hit is going to be the exact same number for both. You only get your Divine strike (I think War domain gets that at 8 and 14 like the Life cleric does) on the first hit so that does favor TWF slightly at 8 and more at 14.
The next thing to look at is the action economy. TWF uses your bonus action every round. Whereas GWF leaves it free. You've got some bonus action spells you might want to use.
Lastly, there is feat selection. A melee cleric is MAD (Multiple Ability Dependent) so you might not want to actually pick up a feat and let both your stats run up to 20, but if you could settle for an 18 str, you might want to pick up the feat related to your fighting style. The GWF feat gives you extra attacks, but the TWF feat lets you wield one handed weapons rather than only light weapons and boosts your AC a bit.
Really, looking at the numbers, the decision is entirely up to you and how you want to play the character, they are quite similar. I give the damage edge to TWF, and the action economy edge to GWF. Ultimately though, at high levels, your cantrips might end up doing more damage than your weapon attacks anyways.
[RPG] Is a Dexterity-based paladin build competitive with a Strength-based paladin as a front-line fighter
Yes, this build is competitive. Compared with a strength based dueling paladin, you'll actually be better off in many ways. Though you can't compete with a great weapon fighting paladin in damage. To illustrate this: let's compare some general build options.
Let's assume there are two paladins: Strong and Dexter. Strong and Dexter have the same stats except that Dexter has dexterity equal to Strong's strength, and strength equal to Strong's dexterity. This means that for the purposes of attack rolls and damage modifiers, the two are exactly the same. For that reason, I'm omitting modifiers when I make damage assessments below. I'm also not including magic items because with enough magic, you can make anything work.
Sword and Board
Dexterity is very competitive here, if not the best choice. In this case, both paladins take the dueling style and wield a weapon in one hand and a shield in the other. The highest damage one handed weapons all deal 1d8 damage. For Dexter, that means using a rapier. For Strong, it could be a longsword, warhammer, battleaxe, or rapier. Regardless, because the damage die is the same, and the paladin's relevant ability score modifiers are the same, their damage output is identical. Eventually, when Strong gets plate armor, they will have 1 higher AC than Dexter does (18+2 vs 17+2, as you have noted) Strong also has options to deal all three types of physical damage (slashing, bludgeoning, and piercing). Dexter will only be able to deal piercing damage, but their initiative, stealth, and dexterity saves will all be higher than Strong's. Dexter also has much better ranged attack options since they're as effective with a bow as with their rapier.
In return for being 5% easier to hit compared to Strong with plate armor, Dexter will be better at range, go earlier in the initiative more often, and make their dexterity saving throws more often than Strong does. Honestly, that's a pretty fair trade off.
Great Weapon Fighting
Dexter can't compete with Strong's damage here. Dexter keeps their shield and rapier. But Strong takes great weapon fighting and a heavy weapon. Their damage die goes up to either 1d12 or 2d6, depending on which weapon they choose. And they get to reroll 1s and 2s on that damage. Using the results of the How much damage does Great Weapon Fighting add on average question, that works out to roughly an extra +1 damage on average. So, strong will be dealing about 8 damage per hit before modifiers regardless of which weapon they choose. Strong could also take the Great Weapon Master (GWM) feat to further increase their damage output. Dexter is still dealing about 4.5 per hit before modifiers with their rapier and doesn't really have any feats that will consistently increase their damage to Strong's level. So, Strong is dealing twice as much damage as Dexter on average. However, Strong loses their shield. So even with plate armor, Dexter now has a 1 AC advantage (17+2 vs 18), making them 5% harder to hit than Strong. And Dexter is still better at initiative, ranged attacks, and dexterity saving throws.
Here's an interesting option. If Dexter has 13 strength, they can take one level of fighter to pick up the two-weapon-fighting style. They also take the Dual wielder feat so they can use two rapiers. Now, Dexter and Strong have the same AC (17+1 vs 18) and Dexter deals an extra 4.5 damage on average with their off hand attack. up until 4th level, Strong will deal about 8 damage per hit and Dexter will deal about 9 damage over two hits. So Dexter hits harder than Strong! As long as both attacks hit anyway.
The issue is that once the paladins hit 5th level they get an extra attack. Strong now deals 16 damage over two hits (8 damage twice), and Dexter deals 13.5 damage over three hits (4.5 damage thrice). And that doesn't factor in the bonus damage that GWM offers Strong. At this point, Strong hits harder than Dexter does, and again Dexter can't do much to catch up.
This is a very quick analysis. Enough to get the point across I think. There is no way for a Paladin using dexterity to deal the same damage as a paladin that's focusing on dealing as much weapon damage as possible. If you want to deal maximum damage, you have to go with a strength paladin. But dexterity gives you much better survivability. Dexterity saving throws are very common, and can deal some really high damage (think fireballs, lightning bolts, and fiery dragon breath). And with higher dexterity you can go earlier in the initiative and position yourself where you need to be as soon as possible. Plus the large damage bursts that paladins are known for because of their smite is, as you have pointed out, not dependent on weapon damage. It's just based on spell slot level. All of that is to say, the survivability and flexibility is something that will definitely serve you well and I personally think is a reasonable trade-off.
The warpriest gets higher-level spells, and is therefore better.
That’s really all it takes. That’s really all there is to say. Barring extremely weird cases where a spell list is very special (3.5’s healer and Pathfinder’s summoner come to mind), looking at the highest-level spell available to a class is a quick way to determine a class’s relative power. Magic is everything in this game.
The best thing about the paladin, from an optimizer’s perspective, is divine grace. That is very good. Might even be worth multiclassing for, in some cases (though probably not as a warpriest since Charisma would not be your first focus). But ultimately, “higher-level spells sooner” is a far more powerful class feature than anything the paladin has—including divine grace.
Beyond divine grace, lay on hands is OK, ish, but easily replaced by a wand of cure light wounds in most situations. In special situations, where the mercies become important, a wand of lesser restoration covers most of it; that’s pricier but will be used less often. These wands are ultimately not quite as good as lay on hands, but they’re close enough that lay on hands is a pretty low-value class feature.
Smite is decent in Pathfinder, but you get so few of them that I’m not super-excited about it. The fact that the warpriest gets several free quickenings a day from fervor easily outpaces smite—you get more uses of fervor, and you can buff yourself with things that lead to similar or better damage than smite does.
And the divine bond comes out to basically “pretty good, but nothing amazing.”
On the warpriest side, you have better spells, and you have the ability to cast them quicker. Fervor is a very, very good class feature. Quicken Spell is a +4-spell-level metamagic for a reason. And the cleric spell list is extremely well-positioned to take advantage of that ability.
Throw in some bonus feats and some free money from the sacred weapon and armor, and you’re in a pretty good spot.
An exception, of sorts
Paladins are in a pretty good place to start working on some Charisma optimization, which is far more “open” to optimization than Wisdom. Divine grace is a big part of that. You can add to that with starknives, which can get Charisma added to attack and damage through Divine Fighting Technique. And there are a number of ways to get Charisma to AC. Having one ability cover attack, damage, AC, and all of your saves is really very nice, and I’m pretty sure you could not do the same with Wisdom.
But there’s a problem with this: that Divine Fighting Technique is associated, in Golarion, with Desna, a CG goddess. Divine Fighting Technique requires that you match your deity’s alignment, and a paladin cannot do that. So you can only do this if playing in a campaign that associates it with an LG deity, or else relaxes alignment requirements in general (e.g. allows a non-LG paladin or Divine Fighting Technique without a matching alignment).
And even when that happens, paladin provides only a foundation; you wouldn’t be staying with the class most likely. Instead, you would most likely be an “oradin” rather than a paladin: taking two levels of paladin for divine grace, and then switching to oracle for one of the Cha-to-AC mysteries and the far-superior spellcasting is strictly superior.