I really want to know that if I take the HD 7870 am I going to see a big difference in video editing like on vegas pro 12 or adobe premiere or should I take the gtx 660 for that matter.
Go to http://www.hwcompare.com/ for comparing the two cards. Power consumption and such, but as for Vegas Pro 12 or Adobe Premiere I have know clue. Hopefully someone more knowledgable will answer you question.
A lot of Adobe programs take advantage of Nvidia's CUDA cores and that will help you out with your editing immensely. From what I've heard, the software developers are working on making AMDs cards perform just as well. Right now, as far as I've read and see, an Intel/Nvidia build is the best for editing.
Well... I used the 660 and I loved it. Never had a 7870 but I've heard good things about it. For gaming and editiong , you'll probably have a pretty beasty rig.
If your editing in Adobes series of programs (I think :S) , they take advantage of the cuda cores which I presume will make it a more joyous experience. For gaming , it sort of depends what games you play; Some games favour Nvidia and some AMD. Personally , I'd get the AMD ,but seeing as your editing and gaming , get the Nvidia option. The 660 is a very capable card. I had the 660 Superclocked by EVGA and it gave me a steady 60fps+ with an i3 2120 on Bf3 on Ultra at 1920x1080 (that was a mouthful :L).
You could spend the extra money on getting a 660 Ti Superclocked and if you ever needed to , upgrade to SLI - which I believe compares to the 680.