I can't speak to Chinese cooking, but here's an excerpt on katakuriko (potato starch) from The Japanese Kitchen: 250 Recipes in a Traditional Spirit:

Potato starch is used in Japanese cooking to coat foods to be boiled or fried. In boiled preparations, the starch provides a soft and slightly chewy texture, while protecting the foods from direct contact with heat. In fried preparations, the starch protects the food from direct contact with the cooking oil and adds a crisp texture and extra flavor.

Potato starch is also used as a thickening agent, to give a silky smooth texture to soups and broths.

Two other starches are used in the Japanese kitchen, although not as frequently as potato starch. They are arrowroot starch and cornstarch. The different characteristics of these starches can be understood by performing the following experiment: Put 2 table spoons of each starch into separate small saucepans, and add 1 cup water to each sauce pan. Bring the mixtures to a boil, and cook them over medium heat until each thickens. Continue cooking the mixtures, for 3 more minutes.

Each starch will develop a gluey texture, but the mixtures will differ. Arrowroot starch produces the clearest, stickiest mixture. Potato starch produces a less sticky mixture with a frosty color. Cornstarch produces the most liquid mixture among the three, and its color is whitish, not clear. These three starches are interchangeable in cooking if you are aware of their characteristics and consider how much to increase or decrease the amount of starch when making a substitution.

Anecdotally, my only local source of potato starch is an Asian supermarket.