What games do you want play (like CS:GO, ARK, COD:BO3, GTAV?), what fps you want reach, what settings are ok (like ps4 quality or best possible ultra settings?)? What hardware do you have? Why do you want upgrade, is the GTX 960 not enough and why not enough? Do you have some problems with it? How much can you spend, maybe there is a better card for you like the GTX 980.
>> Is it worth 60$ to upgrade from a GTX 960 to a GTX 970?
If you talk about selling GTX 960 and adding 60 $ to buy a brand new GTX 970, then yes it probably is worth it. But we don't know what it is worth FOR YOU!
>> How much longer will I be able to use a GTX 970 vs a GTX 960?
Depends on the build quality and how you use them, if you overclock. There is no general rule that a graphics card with better chip will last longer.
>> I'm going to be playing games on a 1080p 60hz TV.
Do you mean 60 Hz you want to reach the 60 fps on pc? Because you can play with 30 fps on 60 Hz monitor or tv too.