The difference is that one lists all posts sorted by time (Recent Posts).
New Posts lists all post sorted by time you haven't seen yet.
So when someone posted 1 second ago something, you will find in both lists this message as the newest one.
Once you have viewed this post, it will disappear from your New Posts list. It won't be anymore the newest in your list, it just won't be there.
But in your Recent Posts list it will stay.
Interestingly on my site it was requested to be added in. Some users prefer to see all recent posts, not just the new ones. They didn't like that Recent Posts was only available to them via the New Posts page (so 2 clicks to see Recent Posts. Now they are happy with the added link and no-one on my site seems confused between the two.
I added mine by editing the template. If you don't want yours and didn't add it manually yourself then you'll have to work out if an add-on did this and sort it that way, otherwise it could reappear if you don't remove it correctly.
If a thread has 3 pages, and 30 posts (10 to a page), and the user has not entered this thread before, then they'll have 30 new posts.
If the user enters/loads page 2 of the thread (eg. search results lead them to a post on page 2) and then leaves the thread, is there;
a) still 20 new posts (page 1 and page 3)
b) still 10 new posts (page 1 is marked/considered read too since these posts are prior to page 2, leaving only the 10 on page 3 as unread)
c) no new posts (page 1 and 3 are marked/considered as read, because the user loaded page 2)
As the user has never entered that thread before, the answer is c).
But imagine he entered the thread before. Let's say 3 pages with 30 posts (10 to a page) but 10 new posts (so last time he entered he read the previous 20).
So there are 10 new posts for him.
Now imagine he enters this thread again, in page 2 (not 3!).
Then the answer would be b). As he is on a page with no new posts, the thread wouldn't be marked as read as the page he entered doesn't own any new posts to him.
So the new posts page would still show that thread.