It's always been possible to run Linux programs under Windows: just run a VM. What Microsoft has done here makes it less painful to run Linux programs, sure, but these programs still exist in their own little world. Cygwin programs, on the other hand, are Windows programs. To me, that makes them much more useful.
Now, maybe I'm wrong. Maybe the new Linux subsystem is more tightly integrated with the rest of the system than I'm guessing. But based on the available documentation, it looks a lot more like SFU or Interix than it does Cygwin, and that's a shame, because if I'm right, Microsoft misunderstood the whole point of Cygwin. Again.