There is a $50 difference in price between these two chips, $390 versus $340, which will be within the price range of many of enthusiasts.  The i7-7700K's cores run at a higher frequency but there are only four whereas the i7-7800X has a half dozen.  The memory configuration is also a factor, with the Skylake chip offering quad channel memory while the Kaby Lake only offers dual channel.  The size of the cache may not have a huge impact on gaming performance but you need to consider the number of PCIe lanes; is 16 sufficient or will you need 28?

Techspot seeks to answer this question with a large number of gaming benchmarks, including PlayerUnknown's Battlegrounds.

"Although we consider the Ryzen 5 1600 to be the sweet spot for building a new high-end gaming rig, many of you interested in going Intel want to know whether it makes more sense to buy the Core i7-7700K or the new 7800X?"

Here are some more Processor articles from around the web:

Processors