Ryzen 5 1600 is similar to Intel's new 6 core i7 7800x in gaming (https://www.youtube.com/watch?v=UfNMn7RWgLw). So I would straight up go for the Ryzen.
6600k isn't a bad CPU either, it is just not as good as the Ryzen 5 because it only has 4 cores. It can make up for the lack of threads with higher clock speeds but that doesn't always work (see minimum frame rates in Metro Last Light), and you will have to always maintain a clean install of windows to avoid hiccups and frame rate dips with 6600k.
If you only game on your computer it would be relatively easy to keep it clean, for example, disable anti virus, remove all bloatware and disable windows indexer(they suddenly decide to run right in the middle of a game and sometimes cause hiccups). However, if you do anything else, then keeping the computer gaming optimised can be a bit of a pain. With the Ryzen 1600 and 1700, you can just forget about background applications.