American exercise gives the buyer an extra right, and any extra right has a price. The difference between an American contract and an otherwise identical European one is the american-premium.
For sellers, the early-exercise right is the source of early-assignment and dividend-risk. Rational holders exercise early only in specific situations — deep in-the-money calls before a large dividend, or deep in-the-money puts when interest on the strike proceeds exceeds remaining extrinsic-value.
Example: XYZ at $50, you are short the $40 call, extrinsic value $0.05, and XYZ goes ex-dividend $0.40 tomorrow. A holder who exercises captures $0.40 and gives up $0.05. Expect to be assigned, and to be short 100 shares plus the dividend obligation.
Related: european-style-option, early-exercise, dividend-risk