"Have to" and "must" usually mean the same thing: there is no choice about something, and you are forced to do it. However, the negatives of these two modals mean different things. "Don't have to" is different from "mustn't":
|Don't have to
||means that something is not necessary, but you can do it if you want to -- it's up to you.
||"You don't have to pass the test."
"It's OK if you pass the test, but it's also OK if you don't pass it."
||means that something is not allowed; it is forbidden, and you definitely shouldn't do it.
||"You mustn't smoke in the classroom."
"Smoking is not allowed in the classroom."