Both verbs refer to necessity. Have to is used when there's an outside obligation or reason:
Must suggests the speaker's strong wish or authority:
For the same reason must is used in written rules, instructions, regulations etc.:
Again, it suggests the author's authority.
Must is also used to describe how something works:
In many situations the difference in meaning isn't very important and both forms can be used. In addition, the past form of both must and have to is had to:
With have to the meaning doesn't change in questions:
But with must the meaning changes in questions - it doesn't make sense to ask questions about our own wishes, we ask about other people's wishes:
Such questions in the second and third person may express annoyance:
When a question refers to regulations or how something works, the meaning doesn't change:
To indicate no necessity or obligation we use don't have to:
But mustn't indicates no persmission:
This meaning is very similar to can't.
We must use have to with the perfect and future tenses, as well as other modal verbs:
Modal verbs can't be used in this way.