Which is correct, "you and I" or "you and me"?

This is an example of hypercorrection, which is when native speakers make an accidental error in their zeal to avoid a different error.

In this case, the error that's being avoided is the error of writing "you and me" in subject position, as in the following sentence:

You and me are going to the store.

This is formally incorrect, although it's very common in contemporary spoken English. Because they have been taught that this is incorrect, many people hypercorrect and change "you and me" to "you and I" in all positions. That is, they incorrectly learn the rule about when to use "you and I", and so produce sentences like the following:

You and I are going to the store. [Correct]

He'll come to the store with you and I. [Incorrect]

"You and I" is the subject. "You and me" is the object.

"You and I hate Phil." "Phil hates you and me." "Phil is hated by you and me." All of these are grammatically correct. (No offence to Phil.)

It really winds me up when people hypercorrect because they think that "you and me" is always wrong. "Phil hates you and I." NO! This is worrying.

The best way to work out if you should be using "you and I" or "you and me" is to take away the "you and" and see if the sentence sounds right with just I or me.

"I hate Phil." Sounds good. "Me hate Phil." Sounds like a caveman talking!

"Phil hates me." Sounds good. "Phil hates I." Sounds ridiculous!

They are synonymous but grammatically different. They are often used interchangeably (incorrectly) by native speakers. There is a simple way to tell which one to use, by imagining the clause without the "you and" part:

Example: You and (I/me?) should spend more time together
Imagine: "Me should..." (this is clearly wrong)
Imagine: "I should..." (correct!)
Answer: You and I should spend more time together

Example: He should have spoken to you and (I/me?).
Imagine: "He should have spoken to me" (correct!)
Imagine: "He should have spoken to I" (wrong)
Answer: He should have spoken to you and me.

An alternative view is that "you and me" was always OK until somebody decided in the 18th century that English should be like Latin and started teaching that you have to use 'I' when you would use 'ego' in Latin. Since the rule taught since then is hard for English speakers to grasp (since grammatical case is marginal in English), many people are unsure of where to apply it, hence the hypercorrection. In fact, in Joseph Emonds' 1985 paper "A grammatically deviant prestige construction" he shows that there are slightly more complicated examples (which I cannot recall to mind) where even people who think they do know the rule are often unsure. His paper argues that English with that rule is not a possible natural language, in the sense that there is not enough information about case available to an English-speaking child to construct the rule. His claim is that the rule exists only as a rule learnt in school or equivalent.