They're exactly the same. "Needn't" is more formal and, like Han Choi said, it's never used. It's one of those phrases that has been phased out due to English's nature to constantly simplify itself.

"don't need to" = "needn't"

No one says needn't. I actually had to Google it because I didn't think it was a word. I can't tell you the difference because the definition of needn't is need not and as I said no one says needn't or need not, but in a sentence I found, it seems like it could be used in place of shouldn't. The sentence is "but your mother needn't have died" or it could be didn't need to die. In which case there is absolutely no difference. :)

These sentences are same but more formal sentence is "Do not need to".....

Ohhh.. Thank you a lot... Now it is clearer :)