The correct answer is the first one: "There's a big age difference between you and me." It's because "between" is a preposition, and after a preposition we use objective pronouns ("me" - also "him", "her" or "us"), not a subjective pronoun ("I" - also "he", "she" or "we").
A good way to figure this out is to replace "I" with another subjective pronoun, and see if if sounds wrong: "There's a big age difference between you and we." Sounds wrong with a subjective pronoun, doesn't it? Then you need the objective pronoun - in this case, "me".
It's a little complicated, but I hope this helps!