"Clamp down" means to impose restrictions or controls. "Crack down" means to take sever measures against some undesirable or (most likely) illegal activities. If I heard that the authorities were clamping down on some activity, I would probably picture some controls implemented. On the other hand, if I heard they were cracking down on some activity, I would expect to hear of arrests.

The differences are subtle and many people use them interchangeably. However, they mean different things. Clamp down - to strictly enforce rules and regulations. Usually not used in legal or police situations Crack down - to take action in a very severe way, often used in legal or police situations. "The company clamped down on overtime abuse.". " The state clamped down on corporate tax loopholes.". "The school clamped down on unauthorized use of its sports facilities." "The police cracked down on rampant drug dealing.". "The military cracked down on terrorist bombings.". You will see people use either term in all of these sentences and it will be understood.