What exactly is a "reap?" I get (or think I get) that it's some sort of trip or throw or combination of the two. But what differentiates it from any other trip or throw (or whatever it really is)?
"Reaping" is a word used to describe the finishing motion of a throw, not a throw itself. In a 'reap" the body part (usually the leg, but can be the hands) carries through as much as possible. It is a very powerful motion.
Think of a scythe or sickle cutting wheat. You bring the blade up high and in an even arc, driving it thorugh the wheat stalks.
The same applies here. I'll use osoto gari ( big outside reap) as an example, as it is one of the more common reaps. This pic: shows an excellent reap here, As White's right leg is very deep on Blue's right. Instead of just putting the foot down behind the opponent's heel (a trip), White leg is almost fully extended, calf to calf, up in the air, continung to cause problems of balance, should the defender be ready. If the defender is ready for a trip, they most likely will not get taken down.
This video does a good job of showing the reap. While the reaping leg does always not look like the above pic, the calf to calf action is present, and never does the thrower put his foot down right behind the opponent's heel- he always drives it back.
Also notice here, though, the tremendous motion of the attacker's upper body- this is more important than the reap itself, as with no offbalance, the throw does not work. For osoto gari, the reap is a mere finishing touch.