I don't rope or go to war on horseback, but I don't hesitate to put a curb on a thick necked pony with a timid young riding student up top. When that kid pulls on the reins, I want that little sucker to stop. Since it has so much more potential for causing pain, thus causing stiffness and resistance, dressage and reining trainers spend the first few years of training without a curb. When the horse is accepting a snaffle bit properly, and is thoroughly trained at all gaits and many figures, the trainer introduces a curb bit. The curb is always used in conjunction with a snaffle for dressage, with the snaffle being used primarily, and for lateral bending, and the curb for additional vertical flexion and elevation.
Horses tend to stretch into contact with a snaffle, but back off of contact with a curb.
Western trainers usually train their horses to perform in a curb only, without a snaffle in the mouth at the same time.