Sometimes they are the same, but there can be a difference. When we say 'needn't have', it means the action was performed but it was not necessary. When we say 'didn't need to', it also means that the action was not necessary but the action perhaps was performed, or perhaps not. e.g. "I spent a long time doing my homework, but I needn't have because the teacher forgot to ask for it" = "I spent a long time doing my homework, but I didn't need to because the teacher forgot to ask for it" Both mean the action was performed but was not necessary. e.g. "I didn't need to work when I was a student because my parents gave me money every month" The action was not necessary and was not performed. Only 'didn't need to' can be used in this way. 'Needn't have' always means that the action was performed.

There is certainly a difference, and this is one of those questions which you find on tests to ascertain your level of grammar. I didn’t need to do any homework yesterday. = No homework was done. I needn’t have done any homework yesterday. = I DID the homework, but it wasn't necessary. "Need" can work as a modal verb, and "needn't have done" shows an imagined past. It indicates that, in reality, the opposite happened.

There is no difference.

The first would be more commonly spoken where I am. Second sounds to my ears just slightly more formal.