Both AMD and nVidia users should see a benefit from switching to Vulkan. Having spent a bit of time earlier looking at the various performance results posted on the Net (tricky given the rampant fanboi s**t) it seems that results are... peculiar. There's definitely an advantage for AMD users (and arguably those with less powerful CPUs). This could well be due to the reduced driver overheads that Vulkan brings to the table. Vulkan also has Async Compute (AC) enabled for GCN architectures and this would likely result in a performance increase (although it's hard to categorically state just how much of an improvement it affords). In the current state, AC is not enabled for nVidia hardware and this will need a bit of tweaking in (I believe) both the DOOM software and nVidia driver (hence I'd expect further Vulkan-based patches in the near future). My expectation is that the improved performance is more down to reductions in overhead rather than AC.
Another interesting thing is that I didn't really notice any difference when I had a quick play with both the OpenGL and Vulkan versions earlier. They both run and look fantastic. So I thought I'd enable the performance metrics and see if I could notice any difference. Well... yes, there is actually quite a substantial difference in some places (and a fairly healthy improvement across the board) if the stats are to be believed. On 1080p I have everything on Ultra (with Shadows and Virtual Texture Pages on 'Nightmare') and it runs a good 20-30fps faster in a lot of places. I even enjoy a maxing out at 200fps in some areas. I'm yet to see it dip below 100fps which, if I'm being honest, is bloody impressive. Knocking it up to 1200p doesn't make much difference to performance as it still whizzes along at an incredible frame rate well north of 100fps. Even at 1440p it's comfortably maintaining 85fps+ without issue with regular visits north of 100fps.
I don't really know where I'm going with this at the moment (!); the benefits reported online seem to vary so wildly. Some nVidia users notice no difference, some have negative scaling issues, some have tremendous improvements. It's hard to say right now what benefit there is and who is likely to benefit from switching to Vulkan. On a separate note I've noticed a couple of issues with the Vulkan rendering (z-fighting and decal issues) but other than that it seems business as usual.
Interestingly it was nVidia showing the benefits of Vulkan over the weeks / months leading up to the release of the [DOOM] game...