I do eat the peels, unless the apple just looks ... well, fake! I don't quite trust them if they look too perfect. But as I mostly avoid those to start with, it's not too much of an issue.

It feels a bit silly not to eat the core, really. You've eaten the rest, why not go all the way? Also, often enough there's really no place right next to you where to throw it away, and going around with a sticky apple core in hand while trying to find a trash can is not my idea of fun... I'd much rather eat it!