They are two different things. Mints are usually hard candies that are used to freshen your breath. Gum can serve the same purpose, as it often also contains mint or mint-like flavor. Gum is just softer and lasts longer, since it is meant to be chewed. Mints will dissolve quickly in your mouth.

Mint can also refer to a flavor (mint-flavored gum) or the leaf that the flavor originates from (The dessert was garnished with a mint leaf.)