Showmanship is judged based on YOUR handling skills and knowledge in the show ring. Showmanship generally has a pattern for you to lead your horse through. Halter is abased solely on how your horse is built. USUALLY there is not pattern for halter classes. The judge will ask you to track your horse them stand in line while he evaluates the build of your horse against the other horses and chooses a winner.
I've only show in a couple local shows, but have only ever used quarters (what we've always called it) in showmanship. In halter, I've always tried to give the judge the least obstructed view possible of my horse.
Showmanship is on the exhibitor, halter is judged on the horse's conformation and overall looks. I believe you use quadrants in showmanship, I think (you're hearing this from a hunter who's watched a few showmanship clinics... lol) the exhibitor uses the quadrants to decide when to step to the side to give the judge the best view of their horse. Here is this years winning run from the AQHYA World show in showmanship.. I heard it was a pretty tough pattern but I'm clueless haha.
Yes...QUARTERS are used in SMS, then are not used in Halter. Halter is based on conformation and the horse's movement at the walk and trot. Showmanship (SMS) is based on the performance of a pattern. This is where you "work" the judge using quarters.