What Is the Difference between a European and an American Style Option in This Context?

A European option can only be exercised on its expiration date, while an American option can be exercised at any time up to and including the expiration date. For a protocol hedging its native token, American options offer more flexibility to sell (exercise the Put) if a sudden, catastrophic price drop occurs before expiration.

However, European options are typically simpler and cheaper to price.