New Posts shows all unread posts. Recent Posts shows all recent posts including those that you have read.
I just looked up the configuration options on this. Our XenForo bbs software keeps track of the last 7 days of recent posts.
"Recent Posts" uses the board's search capabilites to return up to the most recent
200 400 posts, as far back as 7 days.
Out of those most recent 7 days of posts, "New Posts" returns those you have not read, up to the most recent
200 400 of them.
You can prevent forum posts you're not interested in from showing in these search results by selecting the "ignore" option in their forums or threads.
I've just changed the name of the "New Posts" option to "Unread Posts" to better describe its function.
Edit: Search return has been increased from 200 to 400 hits.