Vaporeon's Hydration means that it can work as a pseudo-Manaphy (mainly the Rest-Hydration combo) thanks to Politoed's Drizzle. It's really more of a decision if you want Vaporeon be able to switch into Water moves or not.
Jolteon's Quick Feet is very situational. Even then, you have to give up a useful hold item (like Balloon or Life Orb) to guarantee the speed boost. In most cases, you would want Volt Absorb anyways so you can switch into Electric moves.
Flareon's Guts means that it will achieve 591 Atk the moment it gets a status. However, the only useable status is poison because Flareon gets screwed by its Fire typing and its slow speed. I see it as a high-risk/high-reward sort of scenario. Again, you could also decide that you want Flareon to be able to switch into Fire moves.
Espeon's Magic Mirror is what is putting it in OU at the moment. The ability to reflect status moves and hazards is vastly better than sharing a status effect.
Umbreon has no real reason to use Inner Focus other than to attack through Fake Out and flinch hax.
Leafeon's Chlorophyll makes it a decent physical sweeper in a Sun team. Also Leaf Guard is not as good as it's rain counterpart.
Ice Body is useful for the healing if that's your thing but that's about it. Glaceon is a Hail troll who wants to shoot Blizzards from 130 base Sp.Atk at you with 100% accuracy while dodging your attacks.