> Also, a lot of classical composers enjoyed elaborate key changes: e.g., you'll have your main theme in A major, and then it appears again in D major, and later it comes back as A major. If you have absolute pitch (those composers likely did), then they all sound different, as if you first see a picture in black and white and later it comes back in blue. If you don't, then that part of composition completely passes over your brain.

Even if one doesn't have perfect pitch they'll still notice the key change itself unless it's performed very subtly or after a long pause. If the key change is drastic enough the timbre of the instruments will change as well resulting in a different sound.