Ricers and food mills accomplish the same task, KK. They just do it differently.
A ricer consists of a hopper with a perforated bottom, mounted to a handle. A second handle, with a plunger, is hinged to the hopper. You fill the hopper with potatoes (or whatever), line up the plunger, and squeeze the handles together. Pressure forces the potatoes through the small holes.
Most food mills work through a rotating scraper rather than a plunger. The mill is, essentially, a perforated bowl or pot shaped container (called a screen), often with clip feet for holding onto a bowl. The handle passes through the container, and has a scraper bar on the inside. You load the container, turn the handle, and the bar forces the food through the holes.
Food mills range in complexity from the simplicity of the Foley, to the ornateness of a the Victorinex (sp?) types, which have multiple interchangeable screens (different sized holes), and which can be converted to motorized operation. Some stand mixers, such as the KitchenAid's, have food mill attachments as well.
Is one better than the other? Not hardly. But there are times when one or the other makes more sense. There are two primary differences:
First, a food mill allows the foodstuff to pass through while any debris accumulates in the screen. So, for instance, you can put unpeeled potatoes in the mill. The bar forces the potatoes through the holes, while the skins are left behind. Whole cooked apples can be run through a mill, leaving the skins and seeds behind. Etc. These things would clog a ricer.
Second, volume. Food mills, by and large, have larger capacities than ricers. To make mashed potatoes for a family meal, I reach for the ricer. There's no reason, in my mind, to use a food mill for such a small quanitity, and then have to clean the thing as well. But when making mashed potatoes for a crowd, or when putting up quantities of sauces, and so on, the food mill makes more sense.
There are other advantages to food mills, particularly when canning foods. For instance, I always disliked making blackberry jam, because of the seeds. But with the proper sized screen, the mill lets the pulp through while holding the seeds as trash.
All in all, however, if we're talking about making a choice betweenthe two, I'd say a ricer makes more sense for the average household.