"Servicing" and "routine maintenance" are interchangeable. Every ten thousand miles, for example, you take your car to the mechanic to have things done to the car that will make it last longer. You are going to change the oil, lubricate moving parts, check fluid levels, replace any bulbs that have burned out, etc.
A "check-up", however, is slightly different. If there is a small problem, and you want to find and fix the small problem before it becomes a big problem. This is the sort of thing you would do before a long trip. Perhaps, for example, the mechanic will discover that the car needs new brakes, a tire needs to be replaced, a fanbelt is becoming frayed, etc. On the other hand, it is possible that the car is in excellent condition and the mechanic will not need to do anything to it. (Note that we go to a doctor for a "check-up" not to treat a health problem but to see if we have one that needs treatment.)
If the revisão ordered by the government, it is probably a "smog check" or "safety check". The smog check determines the level of pollution caused by the engine.