Anybody is a little less formal than anyone, and anybody is used more in conversation than in writing.

They mean the same and are interchangeable. "Anyone" seems to be used more often in writing, while "anybody" seems to be more common in speech. There's a theory that "anyone" should be used when talking about a single unspecified person: - Is anyone here the owner of the yellow Honda parked behind my car? And "anybody" should be used for multiple unspecified individuals: - There are donuts in the break room, if anybody would like one. But most native speakers don't know or care about those things, and just use them interchangeably.