Everything Nicholas said is correct, but there are a few additional layers.
100 years ago, "must" was much more common than "have to". Since then, in both England and America, "must" has declined, and "have to" has increased. Personally, I almost never use "must" to express obligation in speech: I *always* use "have to". The same is true for most Americans. In America, "must" is now mostly used to show *inference* instead of *obligation*--we use it to show that we have arrived at a conclusion based on evidence.
EX: "You walked 20 miles!? You must be tired."// "If you did that for him, you must really love him."// "1000 people took that test, and Sam got the highest score. He must be smart, diligent, or both."
There are some relict uses for the old meaning of "must". It appears in technical manuals, rulebooks, and signs.
EX: [sign]"Employees must wash hands before returning to work."//[technical manual]"This machine must be serviced at least once a month, or the warranty will be void."
TL;DR: In modern America, "have to" is almost always a better choice than "must". If you want to say "mustn't", choose "can't" or "shouldn't" or "are forbidden to" or "aren't allowed to" instead.