Need Not Reply

"X Need Not Apply" is a discriminatory message that essentially means "If you're X, don't bother applying because we won't hire you." For example, "No Irish Need Apply" signs were part of a wave of discrimination against Irish-American immigrants.

This need is sometimes called a modal verb (although others find this term inconsistent): it always requires an infinitive without to; it doesn't have do-support in questions and negative sentences; and the third person singular (he/she/it) doesn't have -s. This is the need in your example.

Generally "need not" is less personal and has less to do with the noun "need" than "don't need to". "You needn't be overly concerned" means "It is of no use for you to be overly concerned." The use of modal "need not" makes it clear that the sentence is not about your needs or even about whether something is necessary.

Yes. Say No reply needed. or End of conversation. or Please do not respond.

You're welcome. Go ahead. Be my guest. My pleasure. That's very kind of you. You shouldn't have.

The short answer to your question, though, is that both are correct. I would only add that "need not" is probably less common in colloquial speech, and even then it is often contracted to "needn't".

You needn't worry about anything. They needn't trouble themselves! Jackson had called Consuelo and told her they had plenty of help, so she needn't come. Jackson quickly realized he needn't have worried. Of course, as long as she kept the hood up, he needn't know her hair was a mess.

: something not needed : superfluity purchasing need-nots.