Both will be more than adequate for single monitor 1080p Gaming, I would personally choose the 390(Well 2xFury X now) as I prefer AMDs tipple monitor support, but would recommend the GTX970 to anyone looking to play on a single monitor.
The 3.5/4gb frame buffer thing rarely affects anyone gaming at 1080p, if you were to start playing at 2560x1440 then I would edge towards the 390.
The 390 draws about 50 watts more power than the GTX970 under full load, so if you're using a 500W power supply then I would certainly stick with the GTX970.
The 390 runs significantly hotter than the GTX970, so could create issues in cases with poor airflow (Based on the fact your processor is a 6300 I am making the assumption that your cases isn't a high end air flow optimised case, neat cables can improve air flow marginally [significantly if it has been done really poorly of course]). If you were looking to do an ITX/MATX build when you upgrade your processor, then the GTX970 would be better in a smaller case (Especially if you require a smaller power supply to fit in the case as above).
The processor has no effect on high end GPU choices, Mantle(AMDs API that could leverage unused CPU power) has been dropped by AMD in favour of OpenGL and Direct3D (Not that they ever stopped supporting them, just that they tried to change the market with game developers) so if you choose to go AMD or Intel when you upgrade it will have minimal impact (On this note I would recommend waiting till skylake is released later this year and upgrade to that along with DDR4)
