Both would come after the response to another question, but the difference is extremely subtle. I'm curious to hear others' responses, though I would say that "How about you?" is more likely to come after a "how" question and "What about you?" after a "what" question. I would also venture to say that "How about you?" is slightly more casual and common. But it's so subtle, I think you could get away with either in most situations.
- How are you? Are you sick?
- No, I'm fine. How about you?
- What do you do?
- I'm a teacher. What about you?
- I'm an architect.
In both of these, replacing "What" with "How" or vice versa wouldn't make them really weird. This is just how I would respond as a native speaker.
Or, just dispense with the whole thing and just say "You?" :)