It's not just that AMD's driver overhead is minimized in DX 12 and Vulkan, there is also the benefit of the low level hardware schedulers that asynchronous compute workloads can be implemented to take advantage of. The extra hardware in AMD's forward thinking architecture has taken a long time to pay off, and well, I would say the cost still hasn't been recouped in terms of gaming benefits, but it looks like we are getting there slowly.
From my understanding, NVIDIA's Pascal cards support only a basic level of async compute, which is certainly better than nothing, and at least don't tank in the performance department like Maxwell cards did under asynchronous workloads. AMD on the other hand supports pretty much all aspects of it. A lot of the performance gains AMD is seeing is in the unused resources being better filled with work being tossed into those bubbles of unused resources. It's unfortunate that what we as end users have are simple generic benchmarks, and not the ability to test each card to see how much potential performance is being left on the table due to underutilized resources.
It may be that NVIDIA has a lot of unused resources under DX 11, and simply can't use them all that much better under DX 12. But, without tools to tell us, all we can do is speculate as to what's happening.
However, as the Time Spy benchmark from Futuremark has clearly pointed out, just because software is built from the ground up to be DX 12, doesn't mean it's going to tap into the compute power that's available on the AMD cards either.
And another good example is with DX 11 games such as the Rise of the Tomb Raider and even older games like the Gears of War remake, that get DX 12 bolted on or shoehorned in after release. When the DX 12 patch was released for Tomb Raider, it was actually advised for NVIDIA users to stick with the DX 11 render path, as they lost performance under DX 12, and that includes the Pascal series. Whether that situation has changed yet, I don't know. Gears of War was such a mess at release, it just didn't work right for a lot of AMD users. Now, however, it shines as an example of AMD cards going from worse than to better than equivalent NVIDIA cards.
As more titles are ported from consoles, AMD will have more wins. Until then, NVIDIA is great for DX 11 and OpenGL. They are also fine for DX 12, but don't expect any performance gains now, and especially in the future. NVIDIA has a history of doing far less for owners of previous generation cards than AMD.