First, please allow me to correct your question: 'What is the difference between "What seems to be the problem?" and "What's the problem?" .......... "What's the problem?" is very direct. It implies that there is a problem, and you need to know what it is. "What seems to be the problem?" is what doctors say. It is polite, detached, reassuring, and implies that perhaps there isn't a problem at all.

First one is more polite. Also, it could mean the person answering is not 100% sure. It sounds like something a doctor or mechanic would say. The second is kinda rude in my opinion.