The best thing I have found to answer this question is from One Stop English. Most of the time we use the two togeather and they mean the same thing as TheLoneUnknown stated.
This is what they said about Must and Have to :
1) Must and have to
A key distinction between must and have to can be found in the negative forms. Whereas You must go and You have to go can be regarded as broadly the same in terms of meaning, You mustn’t go and You don’t have to go are quite different, the first indicating that going is prohibited in some way, or even dangerous, while the second implies an absence of obligation or need.
Another difference between must and have to in the sense of obligation can be found in the nature of the obligation. It is possible to say I’m sorry. I can’t come to the meeting tomorrow because I have to go to the dentist at 3 o’clock but not I can’t come to the meeting tomorrow because I must go the dentist at 3 o’clock. On the other hand, if you have a really bad toothache, you would probably say I really must go to the dentist, although have to could replace must in this sentence. A generalized distinction would be that must refers to an internal need or obligation while have to is used to refer to an external need or obligation. It is probably true to say, however, that must can generally be replaced by have to but have to often cannot be replaced by must so in terms of teaching, it is probably a good idea to teach have to for obligation because it is nearly always correct whereas must is often inappropriate. Must and must not are useful for official notices and instructions, e.g. You must carry your passport at all times and You must not smoke in the toilets.