The main difference I've noticed is that Vulkan tends to pause and stutter quite a lot. Though my R9 390X is getting a bit long in the tooth, I should really upgrade it. I like the idea of an API that isn't Direct-X but I like the performance... of the sort of design that tends to cause blue screens...
Oh, and Fraps doesn't work with Vulkan, which is the other reason I switched to DX11.