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?" :)

How about you is used to solicit an opinion from someone. What about you is used to solicit an answer from someone.

How about you? is asked, when someone wants to know how the other person is feeling about something. What about you? is asked, when someone wants to know what the other person is going to do.