This is a tough question, and myself, I'm still trying to "do" dressage using "NH" methods, which are, as far as riding goes, more cowboy than dressage.
The goals might be the same, but the journeys are so very different.
1) NH - cowboy-type riding gives the horse far more freedom to find his balance, as he sees fit,
to do a task or job. As I heard a trainer say once, "He already knows how to . . ." and it was our job to get him to do it.
2) Dressage schooling: the horse is expected to pay the rider far more attention than anything else in the world. He is not
expected to be carrying himself as well as we want, and mostly everything is aimed towards optimizing his movement, as the rider sees fit.
So I guess a lot depends on what you want.
I have a little horse I'm learning sorting on, and while I started her in my normal dressage-way, I find I am now wanting her to learn to turn sharply and side-pass more freely, on her own, without contact. I want her to watch the cows! I'm there for guidance, but I'd like her to do more of the work. (She's got way more innate cow-sense than me anyway.
Where the disciplines meet are in the warmup exercises. They are ridden differently, but leg-yields, circles, transitions etc. are always useful.