Su.Ki.'s answer makes sense, but I would add that in my opinion "must" feels a little bit stronger. At least, "must" sounds more formal and is ordinarily used when the obligation is more serious. For instance, "I have to make a quick phone call / use the restroom / go get my notebook" all sound natural, but "must" would sound awkwardly formal, though not incorrect, in any of these examples. In another example: "I have to be home by midnight" vs. "I must be home by midnight", the requirement sounds stronger in the second sentence. But for sure, the difference is very subtle and you're okay using either most of the time.

Neither of these expressions is stronger than the other. 'Have to' is used more than 'must', because 'must' is rarely used in question forms, and doesn't exist in the continuous and perfect forms. It isn't used in the negative either, as 'must not' is not the negative of 'must'. We make each of them stronger by adding adverbs, such as 'really'.

To add to the answers above, I agree that there is no difference in meaning in most cases. "must" is the better choice in the special case of moral imperatives e.g. "I must become a better person". "You mustn't tell lies."In these cases, we are referring to what is the "right" thing to do. As Su.Ki says, the negative forms have different meanings. Have to = must - obligation Must not - obligation to not do something don't have to = no obligation (you have the choice)

Sometimes there is no difference between "must" and "have to" (I must go. / I have to go.), while in other cases it's important to make a difference between them. We use "must" for expressing personal feelings: You must see this film. (I saw this film earlier and I think it's necessary that you see it too.) We use "have to" for talking about facts: I can't go out in the evening, I have to work tomorrow. I must get up early tomorrow, there are some things I want to arrange. I have to get up early tomorrow, my plane leaves at 6AM. "Must" can also express a strong obligation: You must apologize. (You have no other choice but to apologize.)