The government is the de jure (legally recognized) authority that governs a country (or another political entity).
The regime is the de facto (actual) institution that governs a country (or another political entity). Generally, it is just whichever group of people is currently in control of government. In cases where the government is dysfunctional and ineffective, it might be applied to whichever non-government group has real control.
'Regime' is more often used in a negative sense, to suggest that the government has been hijacked by an illegitimate force. It can be used neutrally as well.