Why would I use Arrowroot instead of Cornstarch?
Compared to corn starch, arrowroot:


Results in a clearer, shinier texture;
Survives the freezing process much better; and
Works better in acidic liquids (certain sauces, soups, etc.)


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.