As stated by Barty1884, I'm pretty sure that all Ryzen 5/7 non-X processors come with a stock cooler, and I'm pretty sure they come with a Wraith Spire/Stealth (the 1600 comes with a non-LED Spire). I'm unsure which processors come with the Wraith Max (which they should be including with the X-based processors since it's a high performance/TDP cooler. >.>)
On the performance aspects of things, there's not much difference. The biggest notable difference between them is the TDP. The 1600 is a 65W TDP, while the 1600X, which is the Ryzen 5 flagship, has a 95W TDP. Same cache sizes, different clock speeds, etc. You get the idea.
If you're just looking at them objectively based on cooling solutions, you'd get the 1600 because it comes with a cooler, while the X doesn't. And, really, you can overclock the 1600 to 1600X levels, so there's really no point in getting the X.
Edit: Doing a bit more research yielded that only the 1800X and 1700X come with the Wraith Max RBG cooler.