What is the difference between a jog and a trot?

The jog and the trot are the same fundamental gait — a two-beat diagonal footfall in which the left front and right hind legs move together as one pair and the right front and left hind legs move together as the other — but they represent different expressions of that gait in terms of speed, collection, energy, and the visual impression they create, and those differences reflect the distinct training philosophies and aesthetic ideals of the disciplines in which each is most commonly seen. The trot in English riding and classical horsemanship is ideally a forward, energetic, ground-covering gait with clear suspension between each diagonal pair's push-off and landing. The English trot at its best demonstrates the impulsion and the energy that collection generates — the horse moving with power from engaged hindquarters, covering ground freely with each stride, and showing the spring and elevation that the training scale's development of impulsion and collection produce. A good working trot in a dressage or hunter context is a gait that clearly covers ground and demonstrates the horse's forward energy and the quality of his movement. The jog in western riding is ideally a slow, quiet, two-beat diagonal gait that is comfortable to sit for extended periods and that demonstrates the horse's willingness to travel at a collected, controlled pace with minimal rein management. The western pleasure jog at its best is slow enough to sit without posting, free enough in the movement to maintain the correct two-beat diagonal footfall, and light enough in the horse's overall carriage to appear effortless. The speed differential between a correct English working trot and a correct western jog is significant — they are both two-beat diagonal gaits but at very different ends of the speed range that the two-beat diagonal footfall can be ridden at. The practical implication of this distinction is that both gaits can become incorrect if they move too far in either direction from their discipline's ideal. A jog that is too slow loses its diagonal footfall and becomes lateral. A trot that is too fast without sufficient engagement becomes flat and loses the suspension that distinguishes a quality trot from running trot. Each gait has a range within which it is correct for its discipline and a quality standard against which it is evaluated that reflects the specific training and movement philosophy of that discipline.