I've heard that arrowroot can be used just like cornstarch as a thickening agent. If I have both ingredients on hand, under what circumstances would I choose one over the other?
Compared to corn starch, arrowroot:
Where it doesn't work so well is in many fruit pies and some other baked goods (because it tends to break down under high heat), and in dairy dishes (you'll end up with a "gooey" texture).
Use arrowroot in place of corn starch whenever your needs match the above.
P.S. Many people are also allergic to corn, and this is probably one of the most common reasons to use arrowroot. If this applies to you, and you're making a dish that is not suited to arrowroot, tapioca flour/starch is another great thickener and actually thickens better than corn starch.
Some people avoid all grains (including corn) due to either intolerance or due to concerns relating to the (negative) nutritional impact of consuming grains OR a philosophical aversion to "neolithic" foods.
For those folks, like those with corn allergies, arrowroot (and other non-grain-based thickeners) may be a viable alternative.