Welcome to the forum, Faemper.
It's too broad a question. It would take too long to give all the details. If you gave some context, it'd be easier.
Basically, must expresses your own conscientious duty and have to - the duty caused by some circumstances. In some contexts it means the person is reluctant to do sth. I must go now. = I think it's right to go now. I have to go now. = I'd like to stay a little more, but I can't be late for work.
Some people make the distinction as Humble pointed out, and some people use them interchangeably (that is, no matter what the circumstances). I believe "must" has a more formal connotation, at least in Texas, because people have a tendency to shorten "have to" to "hafta" when speaking, which is natural but not proper English.