Allow users to choose to backup only specific folders with 1 click, first one a full backup and then subsequent differential backups - BUT allow users to create a new full backup of those folders whenever they want.

There is an option exactly doing that: See their site, and the pictures on the site:

http://www.paragon-software.com/home/systembackup/

Set back up time
Set parameters for snapshot storage
>>>> Set the number of differential snapshots created between two full snapshots
Specify which volumes to secure
Specify exclude masks to prevent backup of unimportant data