Who are you?
This is typically asking for your name, but a particular context could indicate a different meaning.
What are you?
This question is very informal and bit odd. It is a very terse question which implies some understood context.
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If you're in a business meeting it could be your role - e.g. "I'm the accountant"
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If you discussing two sports teams it could mean "Which team are you a fan of?"
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If you're in a group that has been discussing the nationality of different folks in the group then the question would be asking for your nationality.
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If your in a group of college students discussing what year of college you're in then the answer might be "I'm a freshman" or "I'm a sophomore."
So in general I'd say that the question is looking for a label for some kind of subgroup to which you belong.