In affirmative sentences, ‘must’ and ‘have (got) to’ really have the same meaning. But in the negative, their meanings are completely different.

Don’t have to / haven’t got to = you don’t need to / it’s not necessary

• You don’t have to get up early tomorrow. It’s a holiday! (Of course, if you want to get up early, that’s not a problem. You can choose.)
• We don’t have to eat breakfast at home. We can go to a café.

Mustn’t = it’s not okay to do something / it’s a bad idea to do something

• You mustn’t smoke so much! It’s really unhealthy.
• They mustn’t swim in that lake. It’s really dangerous!
• You mustn’t drive so fast!