In many cases they are the same. Buying the book was unnecessary. One major difference is that with 'needn't have' you definitely bought the book. With 'didn't have to' it is possible that you did not buy the book. Examples: I chose the English course at the local college because they supplied everything. I didn't have to buy the book. (You did not buy the book because it was unnecessary.) I bought the book but I wasted my money. I needn't have bought it / I didn't have to buy it because the school supplied it. (You bought the book but it was unnecessary.) 'Didn't have to' is possible in both cases. 'Needn't have' is only possible in the second case.

Take a look at this: https://www.italki.com/post/discussion-216919 The discussion above talks about the difference between 'You needn't have done that' and 'You didn't need to do that', but the distinction is the same. The 'needn't have' construction is actually very useful because it's unambiguous: "You needn't have bought the book" means that it wasn't necessary for you to buy the book, but you bought it anyway. "You didn't need to buy the book" is more common, but - without any context - it is unclear what it means. This sentence can have two completely different meanings. It can mean the same as the first sentence (You BOUGHT the book, although it wasn't necessary for you to do this ) or it can mean that it wasn't necessary to buy the book, so you DIDN'T buy it. I hope that helps.

1) you needn't have bought the book. You had no need to buy the book, it was unnecessary. 2) you didn't have to buy the book. You weren't obligated to buy the book, it wasn't required. In general they are used interchangeably and the form "you didn't have to" is more common. Very minor difference, in most situations there is no difference.

Native US speaker here. I would say the meaning is exactly the same. Needn't is probably a little more common in the UK than the US.