The first sentence implies that he had been drinking and was therefore drunk but does not give any details on whether he was actually drinking in the back of the car or whether he was already drunk when he got into the back of the car. The second sentence implies he was actually drinking in the back of the car. In legal terms this could be a minute but significant difference as to make a false statement would render than statement useless. If the police offer would say in his report that he was drinking in the back of car but this can't be proven then the drunk man might get off. Therefore a police report would state that he was found intoxicated in the rear seat of the car. Unless you meant he was actually in the back of the car, which is what we would call the boot in Ireland.

'Being drunk' refers to the state of being inebriated. This describes the person's condition. 'Drinking' refers to the action of drinking. This says what the person was doing. The second sentence is one which you are reasonably likely to hear. It's similar to 'They were caught smoking at school' or 'She was caught stealing from the supermarket'. The first one is not a normal sentence, because 'being drunk' is not an activity that you can be 'caught doing'. You might say 'He was found drunk in a car park' or 'He was arrested for being drunk and disorderly'.