Both are correct, but "Hvad ønsker du dig" is a phrase that very clearly implies "for Christmas" or "for your birthday". "Hvad ønsker du" is more for everyday life, but more formal and polite than just "Hvad vil du have", and therefore less common. Waiters use this phrase in cafés, being as polite as possible, but they usually use "De" (like the German "Sie") instead of "du".
Unlike German we have more or less outgrown the words "De/Dem", and instead use "du" for everyone whether we know them or not. The only ones using it are waiters, the elder generation and the royal family ;-)
But "Hvad ønsker du dig"? Exclusively for gift-giving.
This phrase litterally means "what do you want as a gift (on the upcoming holiday or special celebration)". In English you wouldn't really use "wish" in this sentence, but rather say "what do you want for...". So in a normal English conversation I would say "want", but if I was translating a text and I wanted to get the point across that I was referring to gift-giving, I would use "wish", even though it's a little unconventional for native English speakers.
So difficult when English is not your mother language. I'm German and would have had any problems to translate into German. My English isn't that bad too. But sometimes I have no idea how to find an English translation. I wrote here "What do you wish for yourself?" and I lost a heart. :(