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Some of these might be in newBB already, I only tested it long enough to see it fell short of my needs (no offense).
Far more extensive permissions, including layered groups.
Private forums unviewable except to those who have perms (meaning noone should even see it is there unless they belong).
Polls in topics.
Subscribe to topics with email digest option.
Very granular permission options.
Integrated and improved private message system.
Mass delete of posts and topics, and ability to delete individual posts.
Auto prune with delete or archive option.
Not sure if this is already there as I havent tested this far - but topic splitting and moving, including option to move single posts.
Better reorganization options (it was easier to edit the db tables than to try and rearrange a large complex forum with the included tools).
Better user ban controls.
Posts included on reply pages so you can easily reference what you are responding to.
File attach options.
Email via board address masking.
Print format view.
Flood control.
Duplicate post detection (for those who are to impatient and just have to click submit twice).
Calendar/event system.
Moderator logging (to keep track of what they are doing) including rollback options to undo mistakes they might make.
RSS options for keeping a close eye on topics or whole categories.
Detailed board statistics.
One click board backups.
PM notification, and read receipts.
One more I just noticed.. After posting.. return to where your post is instead of top of the thread.
*sigh* and another... It will have to be easy to transfer all of the old data to newbb regardless of the size of the forum :)