There are three temperaments, each with their own perks:
Calm (-50 ↔ -20):
Stamina recovery bonus: Horses with a Calm temperament get a bonus to their stamina recovery speed when at rest (not galloping or jumping).
Never rushes jumps
Attempts to slow
Balanced (-19 ↔ 19):
No Bonus (sort of... I discuss this in the Independent Horses & The Cost of Bonuses section)
Sometimes rushes obstacles at 10+
Attempts to slow or speed up towards working (center) speed: If your horse is going at the long canter, they will attempt to slow towards working canter. If your horse is traveling at the short trot, they will attempt to speed up towards working trot.
Forward (20 ↔ 50):
Speed Bonus: Forward horses get a bonus to their speed via faster animation cycles.
Always rushes jumps
Attempts to speed up
As stated above, each temperament gets a bonus (Balanced does not technically have a bonus, but we'll discuss that in the section below). These bonuses scale based upon how extreme the horse's temperament is away from 0. For example, a Calm horse with -50 temperament gets a greater stamina recovery bonus than a Calm horse with -20 temperament. The same goes for Forward Horses. A Forward horse with 50 temperament gets a stronger speed bonus than a Forward horse with 20 temperament.
These bonuses are fantastic, but they do come at a cost. You'll have noticed that horses attempt to adjust speed on their own. The more extreme a horse's temperament is, the more often they will attempt speed adjustments. Horses with more extreme temperaments will also attempt to make this transition slightly faster after the initial signal. The below chart compares the extremes of each temperament:
So, as you can see, while Balanced horses do not get a bonus per se, the closer a horse is to 0 in temperament, the less they will attempt to transition on their own.
There isn't a superior temperament for all occasions! The ideal temperament completely depends on what you're trying to do and what you personally prefer. When I am doing hunter or challenge courses where precision is key, I much prefer riding my Balanced horse. When I am going for speed in competitions, I prefer my Forward horse. We don't yet have cross country, but a Calm horse would be fantastic there due to their stamina recovery potential. Pick the horse that suits you.