Other differences I've noted; feel free to jump in and disagree because they are based on observed generalizations subject to interpretation and also based on what I believe is a fair sampling of courses from each;
I think Tom Doak strives to avoid being pigeonholed or stereotyped by the look of his courses, yet isn't always successful in achieving those course to course distinctions.
On the other hand and as I mentioned on another thread, I think that C&C run the risk of becoming a designer label. In other words, I'm sensing too much of that rough and rugged look whether it translates well and naturally to a site or not. Gil Hanse may be in a similar situation, although his restoration work shows real sensitivity to the original designer and therefore greater variety.
I think that C&C are the best of the bunch at using existing landforms to play into strategies from the tee, usually on a diagonal. Such ploys effectively narrow the optimal target zone, yet remain playable for everyone. Examples include such holes as 1, 16, and 18 at Sand Hills, where a drive landing on the correct (left) side of a slope turbo-boost forward towards the hole while shots drifting to the other side leave one in the fairway but at a disadvantage. They may overdo this, particularly with holes favoring right to left ballflights.
(inside joke to Tom D.)
I think Gil Hanse is the one most apt to introduce humor into a design, sometimes resulting in preposterously odd situations and shots and other times creating something inspired and playfully sublime. C&C by contrast seem rather dour and quite serious-minded. I think Tom Doak probably self-analyzes himself out of some of the more whimsical and happenstance possibilities of his courses.
I think Doak is the most philosophically pragmatic of the bunch, while C&C are the most dogmatic. In other words, Doak will bend to create what he thinks is just another variety of course, or valid design approach, where C&C would probably walk first. It wasn't always that way, though.
None of the group is above creating eye candy, although Doak seems most bothered by it and seems to be looking to revise his bunkering strategies and minimize their use. C&C are probably the most like Fazio in worrying about appearances as a design consideration.
I think it slightly bothers Doak that sometimes his courses are judged as not being true tests for the top 1% of golfers, while I get the sense that C&C could care less and Hanse generally makes his courses plenty hard from the tips, sometimes overly so requiring long carries.
Doak is more apt to use prevailing slopes as a feature of his greens while C&C focus more on creating internal contours. Hanse's focus is more around the edges of his greens.
I'd personally like to see more randomness from all of them.
I would add that they certainly share more similarlities than differences but I'm more apt to have adventurous fun while struggling on a Gil Hanse course, more apt to be in contemplative, cerebral appreciation on a C&C course, and more apt to be intrigued and inspired on a Tom Doak course.