I know I am a month late on this so I hope I am not to late but I will try to save you.
First of all, no. What was said here is NOT correct. There will NOT be a significant gain in performance at 1080p. The GTX 960 is very similar in performance to the HD6970. It is NOT ancient. When the HD6970 was released it was a flagship card. The GTX 960 is a mid range card. They both will give you a fairly similar experience at 1080p.
Don't listen to the people listing benchmarks unless they show benchmarks (non-synthetic) for many games because more often than not they will go find benchmarks for games optimized for the card they prefer and it will be biased. Go do your own research on this stuff and find out for yourself. You can even try typing in something simple into google like "GTX 960 vs HD6970" and almost always get the information you need.
If you NEED an upgrade for your 6970, which at 1080p you probably don't need one, then go look at the 970 or R9 390 (yes, 390 not 290). Those are the latest releases from Nvidia and AMD, respectively, and both will offer much more performance from the 6970.
I hope this was helpful.
Good luck. - See more at:
http://www.tomshardware.co.uk/answers/id-2653477/upgrading-radeon-6970-gtx-960.html#sthash.25c3vdy5.dpuf
My personal experience. Building PC step by step. Now is the turn on VGA. Have hd 6950 2 gb four years old which is weaker than 6970 but still there is no game that I can't play on pretty decent high settings (1920x1080). Like You thinking about upgrade, but 960 is no way to go. It is indeed pretty stronger but as Waughter said You should go r9 390 or gtx 970. This will be my way too.