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Both must and have to can be used to speak about obligation or necessity:

			The doctor says I must stop smoking.
			The doctor says I have (got) to stop smoking.

This is only the case when we are referring to the present or future. When we speak about obligation or necessity in the past, we use have to.

			She had to stop running because she was exhausted.
			I had to pay a parking fine.

It is not possible to use must in this way.

Also, must and have to can both be used to express the idea of certainty:

			He must be the fastest runner in the world.
			He has (got) to be the fastest runner in the world.

To speak about certainty in the past, we use must:

			He must have been late because I saw him miss the train.
			The storm must have spoilt their picnic.

It is not possible to use have to in this way.

When used in the negative, must and have to have different meanings. Must is used to say that something is prohibited, while have to is used to express a lack of obligation:

You mustn't cross the road here. (It is prohibited)
You don't have to cross the road here. (It's not necessary, i.e. you can stay on this side and cross later).

In British English, it is most common to say "have (got) to ", while an American will say "have to ".