The most obvious difference between a fire and a smoke damper is in their names. A fire damper is explicitly used to block and/or resist fire, whereas smoke dampers block or resist smoke.
They also differ in their design
A fire damper usually has a fusible link, which melts when the air in the duct gets too high (as with a fire). The fusible link will then allow the fire damper to shut, cutting off the fire from the rest of the system.
A smoke damper is usually controlled by an actuator that is connected to a smoke detection system. When smoke is detected, the smoke damper’s actuator will receive the signal to close.