The "New Replies" are the new replies to topics you've taken part in (or have pressed the "Notify" button for). For the new replies, even if you log off, and come back in they will stay there until you visit each post.
For the "Unread Posts", the list will clear the next time you log in. So it *should* only show you the new topics since you last logged in. So yes, if you look at the list right now, then log off, then log in again (say) tomorrow then you won't see the list from today. (I'm not sure if that makes sense)
Also, when you are logged in you can browse the board index (you press the Forum button). If you look at the kind of 'badges' they are brown (New) , green (New) or grey (Read).
I've attached a screenshot.
The grey "Read" ones show that there are no updates to that board since you last looked in that board.
The brown "New" badge shows that there is an update in that board since you last had a peek.
The green "New" ones show that there is an update to a sub-board since you last looked, but no update to the parent board. So let's say the "Common Room" has a green badge next to it, it would mean that there is a new post in (say) the "Lighter Side", but not in the Common Room board itself.
Also, each topic you've not looked at since there was a reply will have a "new" flag over it.
Trystan