Between you and me

That's how you say it, and that's how you write it. Between you and I is the result of hypercorrection.

Think about it. You don't (I hope) say, between they, or between we, do you? So why between I? The fact that a "you" is also present in the phrase  should not distract you from the simple fact that between I is both  weird and wrong.

How did this tragic state of affairs come about?

I suspect between you and I is the unforeseen result of tyrannical English teachers traumatizing an entire generation over the misuse of Me as a subject.

Quite rightly, they will have impressed upon you that it is better to say  John and I went out last night and didn't get back until this morning rather than  Me and John or John and me went out last night...

The rule is that I is the subject pronoun  - You say  I went out not Me went out. Therefore, Me and John went out is indubitably wrong.  And yet we use it in speech, or some of us do (I certainly do - often bad grammar sounds friendlier).

Oddly, even the most inveterate user of the Me and John went out formula will never say Me went out (unless the speaker is aged 2).

But between, like all prepositions, takes the object pronouns  me, him, her, us, them  - and of course, you, but you stays the same whether used as a subject or an object pronoun, hence the confusion.