1) The gaming difference for most games would be close to ZERO.
2) Single thread performance is arguably more important. At some points MOST of your processor can be utilized to render but much of the time having four FASTER cores is noticeably better than having six SLOWER cores.
According to Passmark the i7-4790K has a single-thread rating of 2531, and the i7-5820K is 2006. Sure, the i7-5820K can be overclocked as well but let's just stick with these scores as there's no guarantee that the 5820K will do proportionately better (it's lower because six cores add more heat than four).
*Probably it will overclock better (i.e. 4790K might go to 4.6 and the 5820K to 4.2), but that apparently won't close the single thread gap.
The 5820K total score is about 12,997 and the 4790K is 11227 (which means even if the 5820K could be fully utilized it would only benefit by 16% maybe 25% overclocked, though again that's only SOME of the time. The rest of the time the i7-4790K wins)
2) For programs that support GPU acceleration there seems to be a rapidly diminishing return on performance (a GTX980 might give 5% better than a GTX750Ti for example)
Since you're gaming, just go by what you need/want for gaming.
3) System RAM is very important, as is using an SSD properly (main working file should be on it, and any program links for working folders as well.)
*I'd suggest a minimum of 16GB DDR3/4 but it really varies based on file size, program, settings..
4) Budget?
Since you suggest having to keep price down to a GTX980 that indicates a rough budget. Sure, go ahead and get the GTX980 though I'd drop further to a GTX970 if it meant adding a bit more system memory or an SSD.
There's roughly a $130USD price difference there so you could get 32GB instead of 16GB system memory for example, or even put more towards a better monitor. whatever.