5800X boosts a little higher, will transcode with x265 considerably faster, and the extra cores are somewhat relevant for some games.
The issue is the price differential, which is fairly large, and if you're sure you'd use the extra cores on the 5800X, you'd probably be better off with a 5900X, which boosts even higher and has an even larger advantage in cores (vs the 5800X) than the 5800X has over the 5600X, for proportionally less of an increase in price.
I have a 5800X in my multipurpose gaming setup only because the 5900X wasn't readily available when I built it, but if I was on a tighter budget, I'd probably have gone for a 5600X for a pure gaming setup.