The main difference between American or European option is in the exercise ability.
While American options are options that can be exercised at any time upto the expiration date, European options are options that can be exercised only on the expiration date itself. i.e. at a single pre-defined point in time.
Due to difference in ability to exercise, there is flexibility with respect to American options. With a European option, the holder can complete the transaction only at the expiration date. On the other hand, American option allows the flexibility to make the transaction at any point in time prior to expiration.
In addition to the added flexibility, this feature implies that an American option will always be worth at least as much as an otherwise identical European option, and will be worth more in some cases.