It doesn't. WAR is awesome doing lvl 50 content like CM, Prae, AK and Ifrit and Garuda. In those fights nobody cares if you're a WAR or PLD. The WAR class is harder to play and additionally underpowered skill-wise compared to PLD, but with a good WAR playing you won't notice. I run those instances with my WAR because I find the class more fun to play precisely because it's harder and more engaging.
The problem starts with Titan and gets ugly in Coil. The damage you take grows much faster than the damage you dish, which determines your self heals. Titan fights? "Let the pally tank". "But hey, I'm a very good player and have great gear!" "Doesn't matter, he's easier to heal". Or "LF tank for Titan" "Hey, invite me, I'm a war!" "No thx, need pally, and we only take 1 tank".
I've done turn 1 in Coil with several groups and pallies are universally preferred. I have a relic and full DL gear except for 3 slots which are myth ilvl 90 pieces, and have around 7.2k health with defiance and without buffs. Last time I went with my FC and tanked. All night long the healers and dps were whining about how much easier the caduceus fight was with pallies instead of wars as tanks. And truly, we wiped often not because someone f*cked up but because the healers couldn't keep up. Wars don't have the EZ-mode buttons and massive passive mitigation that pallies get. And incoming damage scales even more as you go to turn 2 and beyond. For these reasons I'm levelling up a pally to play endgame content.
DPS is a bit higher too, but nothing to write home about. Theorycrafted damage in tanking stance is 5% - 10% higher with WAR.More seriously though: I'm far from endgame content but I was wondering, do Warriors still do more damage than Paladins even when they have Defiance up and the Paladins have Shield Oath up? Woud be kind of sad if in those conditions Paladins did more damage.