Why does Gilgamesh's casual appearance look different in different routes?
I never play Fate/stay night until the end but know a little, I think it's more like "he has anything in his vault and willing to wear casual clothes, so let's make him stylish" sort kind of thing. Because as I know, other servants don't even wear "normal casual" clothes as much as Gilgamesh, they pretty much always use their astral dress/clothes which is very(?) strong/sturdy (in the case with Saber's dress that can block certain magical attacks).

As for the reason why Gilgamesh wear casual clothes much, I think it's to emphasize his arrogance, I mean he is a king and think himself the most powerful entity in their universe, and mostly because he is ranged-type because most of the time he just throw whatever in his vault, even his strongest weapon Ea isn't melee-type weapon like Lancer's noble phantasm Gae Bolg: barbed spear that pierces with death or even Caster's Rule Breaker or Assassin's Tsubame Gaeshi.