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. source: experience
If affection is maxed out, your Pokemon will occasionally: