Practically speaking, there are three points of difference that I see between a car air purifier and a home air purifier. These are:
Size: You can expect a car air purifier to be more compact in size as compared to a home air purifier as that ensures convenient placement inside a car’s cabin.
Area of coverage: A car’s cabin is quite smaller as compared to a room. Based on that very fact, the efficacy of a car air purifier will vastly vary from that of a home air purifier.
Type of plug: As compared to a conventional Type C or Type D plug in a home air purifier, a car air purifier will draw power from your car’s 12V socket.
Of course there’s the angle of the air purification technology being used.