Seconding rsync. It can very easily only copy what has changed, even if in your case this is new files.

AFP being Apple File Protocol? If the NAS also allows SSH use that, otherwise rsync will happily copy files as long as you can mount the AFP volume on a Linux box. I've never done it, but I'd be very surprised if you can't mount the AFP volume in Linux

Finally, rsync is the quick fix:
 

rsync -arz /src /dest

Only thing to note is to pay attention of trialling slashes on the source and destination. If you leave off the traling slash, rsync will create the last folder of path in the destination, but if you include the trailing slash on the source, rsync will skip that folder and copy subfolders and files of that folder directly to the destination.

Both volumes are formatted HFS+. Doing this on a linux box isn't going to work. Weirdo file and folder names that include eg "/".

HFS+ mounts (readonly) fine on Linux. Read-write is not as easy, but as you're copying data from the volume you shouldn't need to worry, except of course when restoring the backup. Natch.

Forward-alashes, like any other special character (space etc.), shouldn't be a problem as long as you escape then with a backslash.

If and when you replace the NAS, probably best to pick a common filesystem for the volumes.

http://unix.stackexchange.com/questions/118521/copy-files-modified-after-specific-date-using-cp-switches

Is that  supposed to mean naturally?

Natch.

Snatch.

Saying natch makes you guys sound like hipster rednecks.

http://celebrityrepresentation.com/wp-content/uploads/2012/11/MAX1.jpg

Fucking Jethro & the seement pond.

Mister Drysdale!