The distinction is not that huge between the two.
New Posts is a listing of all threads with unread posts. I'm pretty sure this goes back to display all threads with unread posts for the past 10 days.
Today's Posts only shows those threads that have unread posts for the past 24 hours.
Unread Posts only shows the number of new unread posts since your last visit.
Hope that helps, I'm still working on 1st cup as well.