It's important for AMD - their lots of cores at low speed (comparatively) architecture really needed it to be efficient and had fairly long delays in task switching in those cores, they had something stupid like 80% utilization efficiency (as in 20% of cores would be idle at any given time) in DX11 titles.
Async in DX12 fixes a lot of that for them, allowing the true power of the architecture to show itself.

Nvidia on the other hand have gone the route of less cores at higher speed - which means they don't need Asynchronous Shaders to be efficient. The internal scheduler (the gigathread engine in the 9 series) was also fairly efficient (and there was minimal delays in task switching) so Async would not have made a dramatic difference even if they did fully support it.
That efficiency is directly evidenced by power consumption. Performance per watt is a huge indicator of an architecture's overall utilization efficiency.

AMD's cards were always more powerful on paper in the midrange, nvidia released theirs 9xx series almost a year before AMD did theirs 3xx, and AMD priced their entire line up based on those on-paper victories. A 390 should have always beaten a 970 (by a 5-10% margin at that price point) - but it's poor efficiency in DX11 held it back.

Async isn't some magic bullet for free performance no matter what, nor is it an easy switch to flip for a game developer ("It's super hard to tune" - I'd link the source but auto mod will delete if I do). From what I've read it needs to be tuned specifically for the architecture of the specific card families - so probably a game will need separate async tuning for GCN vs polaris.
It requires very specific engine and game design to really benefit from it - honestly Ashes of the Singularity is almost an artificial benchmark for Async. It's a genius piece of guerrilla marketing from AMD (that or just plain lucky if they didn't "encourage" it) - the game looks like ass but still manages to be very taxing on graphics cards in one specific way. A way which allows AMD to dominate benchmarks.

In the end the "winner" (and holy fuck I hate the very idea of winners and losers in hardware generations like it's a fucking sports match between rival teams) will be the company that sells the most cards at the best profit margins.
And in that area AMD have really failed in for the past 10 years, great products, great innovation, great for consumers - hell just overall a great engineering department. But really terrible business management.