I got to choose my Fossil this evening and chose the Sail Fossil, to get Amaura, then I decided to really test this out while training it on Route 7, an easy place for Ice types to battle (and it was my first Ice type!). Here are my results.
Building affection takes time. Pokémon CAN get full, and you need to take the time to entertain and feed them between battles to allow digestion and time to get bored (surprisingly, battling is not a means of enjoyment).
At first, there appears to be no difference in gameplay. Through two hearts of affection, this is indeed the case. However, with the third heart, I noticed the character (Cyclone, obviously) saying to Amaura, "I know you can do it!" Starting here, you see the text at the move selection window say, "Amaura is awaiting Cyclone's directions..." By the fourth heart, the Pokémon clearly looks in the direction of its Trainer, as I saw Amaura looking briefly to the right, then back at its opponent. In the case of a one-hit KO, you might see, "I knew you could do it!" If you get low on HP, the character says something to urge the Pokémon on, something about how it's getting tough but you still know they can do it (I missed the line and am now too strong to really encounter it on that Route, can see later what it says).
After getting the fifth and final heart, thus becoming "friends forever", you start seeing the biggest changes. When Amaura came out for one battle, he was "interested in Cyclone's bag". A couple of battles later, Fake Tears was used and he "reads Cyclone's mind and avoids the move", which also shows a heart popping up (the affection icon) in battle. Two more battles later (he one-hitted a horde of Hoppip in here), he was "bursting with enthusiasm for the battle" with a Smeargle...which again got one-hitted. Later, against Illumise, is was "so in sync that it landed one critical hit", and against a Croagunk were apparently breathing in time with each other or something like that.
So far those are what I've seen. I haven't seen any waking up from sleep (haven't been asleep), but I did see poisoned, which basically incapacitates the Pokémon and prevents playing with it or feeding it until cured. I expect Serebii or Bulbapedia to eventually have a full list of the lines that appear, and I'm sure I'll see more as well. :)
