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.