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.
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.
The one glaring difference between the two is the alignment restriction on the Paladin. A Paladin must be Lawful Good, but a Warpriest can be any alignment. This can open up any number of combat-oriented divine options that just aren't available as a Paladin.
On a related note, the Paladin is a knight in shining armour, a paragon of both law and good. The Warpriest is a more combat-oriented priest, and so can fill a number of different character concepts that just wouldn't be suite by a cleric alone, such as a stealthy worshipper of a god of murder, or a combat-oriented worshipper of a trickster god.
From an optimization point of view, there are significant differences.
Paladin gets different spells to cleric, which are based around shutdowns and immunities and double damage sometimes where cleric gets the raw numbers
Paladin gets an animal companion and a warpriest gets enhancement bonuses to armour and weapon.
Paladin gets Smite a few times per day. Warpriest gets swift action small self-heals.
Overall warpriest is going to have higher atk/ac/damage numbers from cleric buffs, but a paladin will have higher saves and higher burst damage as well as an animal companion/mount that will probably put his damage higher than the warpriest.
For high op purposes, the usefulness probably looks like this -
Normal Cleric >>> Paladin > or = Warpriest.
As far as I can tell, there is never a reason from an optimization perspective to ever take warpriest when you could instead take cleric.