Mustn't is used to express a prohibition or obligation not to do something, such as when a doctor tells a patient they mustn't go to school because they are ill. Don't have to expresses something is not necessary or required, like how students don't have to go to school on holidays since it isn't required. The key difference is mustn't conveys an obligation not to do something, while don't have to means something is unneeded or optional.
2. Mustn’t is the negative of must. We use mustn’t to say that something is the wrong thing to do or when we give someone an obligation not to do something. Doctor: You’re very ill. You mustn’t go to school today. Teacher: Be quiet! You mustn’t talk in the lesson! Parent: You mustn’t stay out after midnight. Friend: You mustn’t use my things without asking me!
3. Don’t / doesn’t have to is the negative of have to. We use don’t / doesn’t have to , to say that something is not necessary. It’s a holiday tomorrow, so we don’t have to go to school. I love Sundays because I don’t have to get up early. You don’t have to wash that shirt – it isn’t dirty. Jim’s parents are rich, so he doesn’t have to worry about money.