They are both right, but the meaning is different.
If you ask "What games do you play aside from Tetris?", the question implies that you already know I play Tetris, and also that I play other games. If I answer "none", I am contradicting the assumption that is in your question.
Asking "What other games do you play..." only implies that you know I play Tetris. If I answer "none", I am giving you information, but I'm not contradicting anything that you have already said.
Questions like "What games do you play aside from Tetris?", where the question implies part of the answer, are called leading questions. This is a fairly innocuous example of a leading question, but some situations (e.g. in a law court) asking leading questions is not permitted - the standard example of such a question being "When did you stop beating your wife".