In this case, they are both correct because you would answer with the same thing: a specific date or day. "date" is a noun, so if you specify that you want a "date" or "day" then you can use "what". However, if you had not used the word "date" or "day" (or some form of those) as the subject of "what" or "when" you would say "when" only. For example, you could also say "When is the form due?" but not "What is the form due?" You could say "What is the baby's birthday?" or "When is the baby's birthday?" However, you would not say "What is the baby's birthday party?" because "party" is the subject of "what" and "birthday" is an adjective to "party" in this case. Instead you would only say "When is the baby's birthday party?"