Amy asks via Twitter:
The quick answer is that a ricer is not doing as much to the potato as the mixer is. A ricer uses a lever to force the potato through small holes, which quickly separates the potato into smaller parts. It's kind of like part of a Play-Doh playlet extruder, except the potato isn't quite as flexible as the Play-Doh, so it gets broken up into smaller pieces. The hand mixer applies a lot more force repeatedly on the potato, which breaks apart molecules which stuck together during the cooking process. Whatever you're mixing in will either get in-between relatively few of the molecules, in the case of the ricer, and significantly more of the molecules in the case of the mixer.
We could explore this topic in significant depth if you are really, really interested. Whole books have been written on the effect of various cooking and processing methods have on different varieties and types of potatoes. This has been a topic of interest to food scientists and their predecessors for centuries, because lots of people enjoy potatoes and their versatility. If you'll forgive me, I'm going to gloss over most of the details and just go with the highlights.
Potatoes, when cooked, have two major sets of molecules that affect texture: starches and pectins. Starches, when cooked in liquid (often the liquid within the potato itself, not just boiling or similar), will absorb that liquid and expand tremendously once the starches and water hit the right temperature. We call this gelatinization, and we take advantage of this trait to thicken up our soups and puddings, as well as to make our bread crumb tasty and shiny.
Pectins are the molecule responsible for firming up jellies, jams, and marmalades. In the case of potatoes, the pectin doesn't gelatinize, but it does do some stabilization and emulsification, which helps the cream of butter you add to the potatoes hold together a bit better than it otherwise might.
So, you've cooked your potatoes, and you have all of these starches that are either in short of long chains, depending on whether the potatoes are waxy or mealy, respectively. When you use the ricer on the potatoes, you are allowing the gelatinized starches to remain relatively cohesive, so that your cream, butter, sour cream, or whatever other tasty additions go into your potatoes are surrounding the clumps of starched potatoes. When you use the mixer, you are doing a more thorough mixture of the liquid and the starch, so it's more like a sauce than the riced method. Also, the riced potatoes will have inconsistently mixed pockets of air based on what bubbles haven't collapsed after adding the other ingredients; the mixed potatoes will have air more evenly distributed throughout the mixture from the mixing process.