Short answer: No difference. In my head, "needn't have" sounds a little more British, or sophisticated. I live in Indiana and don't think I have ever heard anyone say "needn't have" in real life. "Didn't need to" is absolutely the more common phrase here. The reasoning, as far as I can guess, is that when you unbuckle the contraction you are left with "need not" and "did not." I can't imagine anyone outside of a Thomas Hardy novel saying "He need not have gone to the farm." (My grandparents are farmers and *definitely* would never say this, haha). You also get a little more accent on the "have" in "didn't have." Think of a whiny child. "But he didn't haaaaaaaave to go to the farm." If you rearrange the words a little, you have two practically synonymous sentences: "You didn't need to [pay for dinner]" and "You didn't have to [pay for dinner]." One of these options is what I would suggest. "Needn't have" is highly uncommon, at least in this part of the country. But like I said, at the end of the day, both make sense. One just sounds a little less colloquial.

In my social circle, "You needn't have (done X), but thank you" is a polite formula. For example, a dinner guest brings flowers and the host says "You needn't have brought the flowers, but thank you. They are charming." "You didn't need to (do X), but thank you" is an equivalent polite formula. However, "You didn't need to (do X). You should have (done nothing or done Y) instead." is a criticism. It's equivalent to "You should not have done X, you should have done Y."

In the English language, particularly in America, you rarely hear people say "needn't have". It is a form of saying something, but many people, even here, would get confused with something like this. So here are some examples: 1. Annie didn't need to study over the weekend because she did not have any upcoming exams. 2. In the end, Annie didn’t have any exams this week, so she needn’t have studied.