We use mustn’t when something should definitely not be done.
- I mustn’t forget to do my homework.
(Definitely don’t forget to do your homework.)
- You mustn’t walk on the road.
We use needn’t when you have a choice to do something or not.
- I needn’t do my homework tonight because the lesson is next week.
(You can do your homework, or not. Your choice)
- You needn’t bring some water because I have some.