RoboKaren's answer is great on the pros and cons, so I'll restrict myself to explicitly answering
... why would someone get the seatpost-mounted rack over the frame-mounted rack?
Really, the only reason is that your bike won't support a frame-mounted rack: for example because it has rear suspension. In engineering terms, a frame-mounted rack supports the weight of what you're carrying from below by being attached to the bike somewhere near the rear axle, and the connection near the saddle is mostly to keep the rack in the correct orientation. A seat-post-mounted rack supports the weight entirely by torsional forces through the seat-post, which isn't nearly as sturdy. That means the rack itself is probably heavier and probably can't support as much load. So you'd almost always want a frame-mounted rack if that's compatible with your bike.
Actually, there is one pro of seat-post-mounted racks that RoboKaren doesn't mention. If you're not carrying much weight, a seat-post rack with a small bag on top of it is more aerodynamic than carrying the same load in a pannier on the side of a frame-mount rack. (Actually, GCN's test was with a large saddle bag, rather than the exact situation I describe, but that's basically the same thing.)