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.
A hunter pace is run cross country over terrain and obstacles similiar to what's found in the hunt field. The hunt in our area provides riders with a description of a fictitious hunt, ("Hounds opened in Cooper's Bottom, and ran a four mile point across the Morehead dairy property, and put their fox to ground at White Gate." so it's also a test of the participant's knowledge of the territory, foxhunting terms and customs, and most importantly, pace, hence the name.
Prizes are given for closest to ideal time. There are also checkpoints you must pass through and collect a poker chip to prove you rode the whole course. Prizes may also be given for fastest time. (At our local hunt, it's a bottle of liquor for the fast time, ribbons and bragging rights for the ideal time.)
They are a boat load of fun, and usually require a partner or team, and a fit horse.
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.
A hunter derby is more challenging than a standard hunter over fences class, require a higher level of horsemanship and a bold, adjustable and adaptable horse.
The format usually has three judges posted a various points of the cours, and is scored numerically, with the three scores being averaged.
This format is also sometimes called a "Hunter Classic."
Go to YouTube and search on "Hunter Derby" and watch some video, it'll give you a good idea of what's expected. There's some high dollar/high prize money ones out there, but there may also be Derby classes at your local show circuit.
I highly recommend you do both! Great fun both of them.
ETA: forget to mention - the Derby format usually has the top of the class return for a second, shorter and more difficult round over fences. And the top of the class may be brought back to hack off to determine final placings.