The GTX 970 is certainly a better card than the R9 380 but the I5 6500 destroys the FX 8350. Right now, in gaming, the 970 with 8350 would probably be slightly better just because the 970 is a beast.
But if I were you I would personally go for the i5 6500 and 380 for a few reasons:
The FX 8350 is 4-5 years old now
The 8350 has terrible single core performance compared to the 6500, which results in the 6500 beating out the 8350 in just about every title, with CPU intensive titles showing the i5 beating it out by like 30-50 fps.
If in a few years you want to upgrade the GPU the 8350 won't be able to handle it and bottleneck it, which it does already to the 970. Whereas the i5 6500 would handle any gpu for quite a few years to come.
The i5 supports DDR4 which supports higher frequency RAM that gives better gaming performance and provides far greater future proof-'ness'
The i5 runs cooler, quiter and draws less power.
Btw I am a FX 8350 owner, and my friend has the same GPU as me but he has a 4690k which outperforms my system in every title.
I recently tried a i5 3570k which is quite a few years old in my system and I saw a massive increase in performance. Especially with titles such as GTA 5 in which I saw a 30 FPS boost. In older titles in which the performance was closer, I saw the i5 keeping far more consistent frames whereas the 8350 would dip and stutter a lot.
People tend to upgrade their GPU's more often then their CPU's as with CPU's you need a new board etc. There are people still running first gen i7's as they just don't need to upgrade yet as they are still handling high end GPU's fine!
I'd say get the best possible CPU you can afford.
Side note: If you want the best i5 for gaming, I'd go for a 6600k as it is overclockable which you can do 2-3 years down the track (which would put it back on par with the new CPUs that will come out 2-3 years from now)