I know my Sylveon dodges moves and gets crits more often. It also gains boosted EXP but I'm not sure that's from Amie or not. It's not a traded Pokemon so its possible that its one of the boost from Pokemon-Amie but I'm not sure.
Yes the dodging, crits, and EXP boosts are from amie. Those are really the only befits besides the happiness boosts.
If affection is maxed out, your Pokemon will occasionally: