I'm not a native speaker in Korean, but I just searched on 다음사전. It seems that 자동차 means car (just any car), and 자가용차 is one's own car (used for personal issues). So if you went to an automobile museum, you would be looking at 자동차, not 자가용차. If you drove from 서울 to 부산, then it would be either one. 자동차 is much more common than 자가용차. 차 is much more common than 자동차.