What are some symptoms of a broken hand compared to a sprain without a broken bone?

Often the same. Symptoms of hand fracture and sprain include pain, swelling and decreased motion. Hand fractures and sprains can occur from similar mechanisms of injury. Moreover, what looks like a simple sprain can be a complex fracture that requires treatment (immobilization or surgery). Therefore, it is important to be evaluated if symptoms from hand/finger injuries don't improve after a few days.
Intensity/duration. Broken bone often but not always associated with more intense pain intially. Broken bone will aften preclude ability to weight bear through affected body part. Displaced fractures (breaks) will make affected body part appear deformed (shortened, rotated or angulated finger for example). Pain from broken bone may persist longer than sprain.