Both cards are pretty decent, so it's not a easy choice. Both cards are very fast gaming cards.
Nvidia has 3d support (with the glasses and with certain monitors)
physx and cuda. They have more limited open cl than amd.
Amd has better open cl support. But no physx, cuda or nvidia 3d support at all. It seems like amd's open cl is going to become the next "x86". This maybe matters more for slower laptop apus that need this help more. Most faster clocked desktop cpus or apus don't need the help of open cl.
Amd also has their free video converter, avio video upscaling software, and "hydravision" that lets you have multiple desktops with fast switching between them with a hotkey.
With hydravision you don't need to run two monitors to do the tasks you mentioned. It gives you up to 9 desktops, but running 2 or 3 is more useful most of the time.
Nvidia also has this, but i have never tried the nvidia one. It's called multidesk.
Both cards are pretty close in speed, i would pick the one that has the features that you want.
Don't expect the already factory overclocked 670 to overclock even more, i kinda doubt it will.
The hd-7970 should overclock a bit, but most card makers are now cherry picking the better gpus for their oc editions. As a result the non oc edition cards don't overclock so well.
I would judge them at the speeds they ship at, and any overclocking you get out of them is a bonus.
Look at benchmarks for the games you like to play first. Some games favor either amd or nvidia cards. If most of your games favor one card, then that is the one to go with.