So, would things like Photoshop (and using onOne plugins) or video processing e.g Handbrake be helped?
no not really
video memory apart from being buffer to contain the MB required for displaying the image on the monitor
is only used as texture buffer for openGL and directX if you use windows
and only if the 3d program or game use large enough textures for their polygons
photoshop and handbrake being pure 2d apps does not benefit from more video memory
There are more than just memory differences in the two cards. When I researched it the 680 had almost double the performance of the 675. That's more than just memory differences and while conventioanl thinking would say the difference for 2d wouldn't be noticable applications that make use of core image use the video card to offload a lot of opperations so I would guess you will notice a difference but if you didn't know what you were missing you probably wouldn't.
I found some of my notes and this should help enumerate the actual hardware diffs for you and they are pretty significant.
CUDA Cores: 1536
Fillrate: 92 billion/s
CUDA Cores: 960
Fillrate: 48 billion/s