I received an interesting question from an English teacher in Iran who wanted to know the differences, if any, between "must" and "have to/has to." This is an interesting question because the expressions are nearly identical. To answer, I had to think not only about their strict definitions but also about how they are used.
The terms "must" and "have to/has to" are modal auxiliaries that communicate (1) an obligation to perform some action or (2) that some state of being or action is highly likely. They are nearly interchangeable.
These expressions have differing connotations when used to communicate an obligation. Unlike "have to/has to," "must" communicates a sense of moral obligation such that the action is the morally correct action. They also have slightly different connotations when used to express states of being or the likelihood of an action. Unlike "must," "has to/have to" can describe the absence of any other choices.
Finally, "must" may express an opinion as a moral judgment, as in, "In my opinion, this action I am telling to you perform is the right action. If you do something else than what I am telling you, you are doing something bad." Thus, similar to "should," "must" can take an opinion and change it into a moral judgment. This is true because morals are value judgments, which are subjective.
In some contexts, "must" is similar to "should" in that "must" communicates a sense of moral obligation to perform some action or moral correctness of a choice. In other contexts, "must" communicates that a certain state of being is so likely that any other possibility is inconceivable.
In most contexts, "has to/have to" communicates a requirement to perform some action, similar to "must" but without the sense of moral correctness. In other contexts, "has to/have to" communicates that a certain state of being is so likely that any other possibility is inconceivable.
"She has to pay her bills on time." –obligation to perform some action.
However, as mentioned, in casual speech these words are often interchanged. A particularly astute person can draw on the connotative differences to argue against a statement that he "must" or "has to" perform some action. A person may use one connotation of the expression, and another person can argue using a differing connotation. This will create conflict. If it doesn't create conflict, it will, at least, create confusion.
For example, if my wife says, "You have to take out the trash," I could respond, "I could watch TV instead." Here, she is using the connotation that expresses an obligation, but I respond as if she meant that no other options are possible. I reply that her statement is wrong because I do have other choices.
On the other hand, if my wife says, "You must take out the trash," I could respond, "No, I probably won't." Again, she is expressing an obligation using words that can communicate an opinion about the morally correct choice. I reply as if she meant that taking out the trash is the most likely, or highly probable, action, which, in my opinion, it isn't.
In practice, neither response will be well received. My response is pretty much guaranteed to create conflict.
The best communicator will seek to understand what the other person intends to communicate with "have to/has to" and "must," and then will respond to that intended meaning, regardless of the words used.