In addition to what Erik wrote, another big difference is in the materials. 2-piece balls typically have a cover made out of Surlyn, a plastic blend that is very durable, and a core made out of rubber combined with filler (usually something like tungsten) to attain the proper weight. The Surlyn cover is what makes a lot of 2-piece balls feel hard and spin less, though newer blends of the material are softer and spin a little more. But that's why these balls are often called "distance rocks": they go a long way, but they often feel hard and they don't spin much on the greens.
A 3-piece (or 4-piece) ball usually has a softer cover made of urethane. Under the cover is a "mantle layer" of Surlyn, then the rubber core (and in the case of a 4-piece ball, a smaller core within the core). At high swing speeds (driver, long irons) the ball compresses enough that the hard Surlyn mantle layer compresses and snaps back for more distance. At slower speeds, the soft urethane cover stays on the club a millisecond longer and spins more. And the cover gives the ball a much softer feel than a 2-piece ball on short shots and putts.
Some companies are now making 3-piece distance balls (HX Hot is an example) that have a softer Surlyn cover and a harder Surlyn mantle layer. The idea is to get all the distance without the hard feel. They succeed to a degree, but don't confuse those with a high-performance 3-piece ball, like a Pro V1 or a HX Tour.
OK, that's not the 3-sentence version. But I hope it helps.