Yes, Copywhiz can just copy new and modified files to the backup folder.
Watch this short video that perfectly depicts this feature:
After you run Copywhiz , click on Advance Settings button and choose "Paste only new or modified files" option.
First time, it will copy all the files since it wouldn't find the destination folder. From second time onwards it will only copy new or modified files.
You can download and install Copywhiz from our website http://www.conceptworld.com/Copywhiz/Download