Is this answer redundant? Let's try anyway, since the implied message of the two is interesting.
"She need not" is an older phrasing for "She does not need to", it's phrased to imply not needing to do something. It's popular in sales and advice, partly because it gives the client a feeling similar to getting something for nothing, I think.
"She needs to not" is an active instruction to stop, which has two implications. Using "need" gives a more commanding tone, saying you "need to do this" rather than "should do this". Combined with "to not" I'd see it as implying the person already doing the action they shouldn't or may to start doing it. It's not as bad an explicit "Stop [...]" but some people can react negatively if they don't like the implied behaviour. I find that generally a qualifier gets attached ("When she gets there, she needs to not ...") and/or rephrased to a "better" action ("remain calm" rather than "not worry").
To contrast with a slightly different action:
- "She need not panic" will hopefully be interpreted as "there's nothing you'll need to panic about".
- "She needs to not panic" might be interpreted as "You're panicking and need to stop".
Perhaps you've heard an advert use the words "You need not delay"? Sneaky, right? If they say
You should buy this now!
they'd sound unpleasant and you might not be as accepting of their claims. But if they instead phrase it to something commanding, like
You need not delay, our operators are standing by to take your order.
then they're implying that you have no reason to not buy it immediately, without actually telling you that you should.
Another one might be changing "You don't need to worry about" to "You need not worry", as a way to convince you that there isn't something you definitely should worry about. This kind of phrasing isn't always reason to be suspicious, but it is certainly understandable why a lot of stereotype dodgy salesmen characters use it while trying to sell their miracle product, before disappearing in the next scene.