They're extremely similar, and mostly interchangeable, although "I don't have tea" suggests you may have some other drink (such as coffee). There also is a third phrasing, which is even more frequently used, "I don't have any tea". A fourth phrase is old-fashioned in British English and extremely old-fashioned in American: "I haven't any tea".

Are they BOTH correct? Yes. Some might prefer: I don't have any tea.

I think "I have no tea" shows that specific time when asked about tea And "I don't have tea" shows that usually having no tea I think like this but answer can be different