"Which is correct: between you and me or between you and I?"

The problem here is that any phrase ending in ...and me has a stigma attached to it because of how it's often misused in one kind of English phrase. In "non-standard English," that is, in many types of informal or colloquial speech, when the subject of a sentence is anybody else and I, me is often used after and even before the and; for example, Billy and me went down to the park, Me and my brother did it. Most speakers of English are taught that I is correct and me is incorrect in sentences like these because I is the standard subject pronoun.