In hunters the idea is to show how quiet and scopy the horse is. In the derbies they like the bigger flowing gallop
I do hunter pace regularly and ride with a recognized hunt that is one of the most aggressive hunts in the area. We ride actual foxhunt territory and many of our jumps are actual stone walls and fencing into and out of pastures (all jumps are optional with go-rounds or round-a-bout routes for non jumping riders). The way our pace does it is in three divisions : open, novice, and hilltopping.
A hunter pace is an event for actual foxhunting horses, and is usually sponsored by a hunt club, though some are open to anyone to participate and it's a run cross country over terrain and obstacles similiar to what's found in the hunt field.
A hunter derby is a show hunter competition, usually a year end or championship event. Some require that you qualify to compete. Hunter derbies differ from a regular hunter class at a show in that the course is usually longer - 12 to 16 fences as opposed to the usual 8. They usually include a hand gallop portion, and may include a narrow fence and a trot fence. The course itself has a mix of related and unrelated distances and an emphasis on natural obstacles, a big change from the standard outside line/diagnol line type of course.
Hope it helped you :)