I've always thought of "happiness", as in the kind that boosts Return, to be a measure of the Pokemon's trust in your skills as a Trainer as well as its gauge of how much you trust it. So for example, if you let a Pokemon faint in battle a lot, it will come to doubt your ability, hence its happiness going down. Conversely, when it gains a level, that's because you were able to bring out its potential, hence happiness going up. Using a Pokemon in an important battle also raises happiness because it conveys the idea that you believe in it and can depend on it for the big battles.
Amie affection is more just general love; that is, how much the Pokemon enjoys being with you. Showing it love and care in Amie (read: rubbing it at speeds that could cause friction burns and force-feeding it cupcakes) does exactly what the game says: makes it more affectionate towards you, because you're showing affection for it. Thus it doesn't want you to be disappointed in it / potentially box it, so it works extra hard in battle, thus the free Shed Skin, Focus Band, Brightpowder, and Scope Lens.