You title it budget gaming and still thinking.... 960 is the budget card . 970 is borderline enthusiast grade.
If you buy a 970 your frames will be over 90fps in the most extreme games like battlefield 4 , which is a very good figure , but you are playing a game not counting frame. Wont matter if it is 70 or 90 (70 with a gtx 960). A 970 will serve you for a longer time but not really long , since 4k gaming needs a technological innovation to be cheap and not much innovation is done in that part , so an innovation down the road ahead in 2-3 years is quite likely. After every 3 years the top end graphics card is surpassed by an entry level ddr5 . (comparing gtx 680 with gtx 960). So i guess Switching more frequently will serve you better for the entire duration. Compare buying 3 gfx in 9 years for 250$ each and 2 gfx in 8 years 450$ each. And see the trend of power gap between each generations. you will just be paying more for higher frames and visual quality which are not visible in FULLHD. For 4k , no doubt you need a 970 .
P.s. Everyone has his own budget , find how much you are willing to spend by imagining your bank balance - the card cost. The only reason someone hasn't suggested you a 980ti is because you asked between 970 and 960