Hi Joanna. Needn't is short for 'Need not' - 'You need not do that'. It's another way of saying 'You don't need to do that'. 'Don't need' would be used like this: 'No thanks, I don't need that.' 'I don't need to go shopping, I went shopping yesterday.'

Honestly, as a native English speaker, I don't think I've ever said "needn't." To me it seems like a contraction that shouldn't be. However, it is apparently taught to English learners. You may try this quiz to check that you know the difference http://www.usingenglish.com/handouts/108.html

I wonder, whether it is correct to say, that "needn't (do)" tends to be a traditional way of saying, while "do not need (to do)" tends to be like American English. [Native speakers, please kindly rectify it should the above presumption be wrong.] Nevertheless, I personally would simply use "do not need (sth. or inf.)" to express various situations. For instant, I need a car. (affirmative) / I do not need any car. (negative) I need to write a letter. (affirmative) / I do not need to write any letter. (negative) Good day and ciao.