Simon has already explained the difference between ‘needn’t’ [need not] and ‘oughtn’t’ [ought not]. However I think that the contraction ‘oughtn’t’ sounds slightly strange, and that we would more likely say ‘ought not’, although the others - ‘mustn’t’ and ‘needn’t’ - sound natural. Another variant is ‘shouldn’t’ [should not] which is equivalent to ‘ought not’, and probably more commonly used.

I would use OUGHT when talking about something that should have been done but wasn't. So it expresses regret or things that weren't done correctly/ I ought to have got up earlier today to have seen the sunrise. So invert this advice for OUGHT NOT. Need is used for necessity, so NEEDN'T is used when there is an absence of necessity. I have a dishwasher, so we needn't do the dishes by hand. I didn't need to set my alarm as I woke up anyway

I think the others covered it pretty well. FYI, ss a native US speaker, we would say SHOULDN'T rather than OUGHTN"T I've only heard the word OUGHTN'T used once, and that was in a country song from 1949.

need not: It is not necessary to do that, but no rules are broken if you do. ought not: You should not do that, for you will be doing something bad if you do.

They are synonymous. You can use both interchangeably