While a food mill or ricer is great for starchy Russets (where the goal is to keep the "mealiness" intact for a fluffy result) the tamis is designed for refining the potato and making it optimal to integrate into a fat-in-water emulsion. 

Tamis

The science of the tamis lies in its high-shear refinement. When you push a simmered waxy or "all purpose" potato (like a Yukon gold) while it's still very hot through the fine mesh of the tamis, you are essentially "atomizing" the potato flesh. Because waxy potatoes have higher pectin levels and smaller starch granules than starchy ones, they can withstand this intense mechanical work without immediately disintegrating into a grainy mess. However, this process also releases a significant amount of intracellular starch. By adding your warm cream and butter immediately after processing the potatoes, you immediately coat these starch molecules. This prevents the starch from forming the long, elastic chains that create a "gluey" texture you might get from over-working Russet potatoes. So the result is a silky mouthfeel in your purée. 

Beyond this use, the tamis is a great tool for homogenization and aeration for pastry and sauces. It's a great refinement tool anywhere that we want a more refined output than you can get through whisking without the overprocessing you might get from an immersion blender or other intense shearing tool. Examples might be a fruit coulis or a custard.

In the world of "modernist" starches, it is also used to create ultra-fine vegetable "flours" or to sift dry ingredients like cocoa or nut meals into a state of total fluffiness. In every application, the tamis serves as the final arbiter of texture, ensuring that the finished product has no "topography" on the palate—just a perfectly smooth, continuous transition of flavor. If you're making something from almond "flour," passing through a tamis will give you just the smallest "grains," which you can use for something else, so if you make macarons, for example, they will be very smooth.

Food Mill

A food mill uses a slanted blade to create a "wedge" of pressure against a perforated plate. As you turn the crank, the blade forces the soft pulp through the holes while the scraping action keeps the larger, tougher fibers (tomato skins, grape seeds, potato skin, etc.) trapped above the plate.

The key here is selective filtration. Because the mill does not use the high-shear scraping of a tamis, it preserves much of the cellular integrity of the food. This makes it great for high-volume tasks where you need a uniform texture but want to avoid the gluey results of a blender or processor. It's the go-to tool for a pomodoro or applesauce, because it incorporates just enough air to create a light, rustic body without emulsifying the natural oils into the water phase, which would turn a bright red sauce orange.

Ricer

The ricer is a simpler mechanical device that functions through extrusion. By using a plunger to force cooked food through a matrix of small holes, it separates the food into distinct "grains" (thus the name). Unlike the food mill or tamis, there is no lateral or circular motion — only downward pressure.

From a scientific perspective, the ricer is the gentlest way to process a starchy vegetable. When using a mealy Russet potato, the ricer allows the large, starch-filled cells to separate from one another along their natural boundaries (the middle lamella) without rupturing the cell walls. Because the cells remain intact, the starch stays "locked" inside, resulting in a mash that is incredibly fluffy and light rather than dense or sticky. This is why a ricer is the best tool for gnocchi. It ensures the potato remains dry and granular, allowing it to incorporate flour without becoming gummy.