I think it depends on the type of car, what your insurance covers, and what risk you are willing to take.
I have used Road Scholar Transport, and Intercity Lines.
Both of the truck trailers were enclosed (not with canvas), and were specifically suited for transporting antique/classic/exotic cars.
Both of those, and a handful of others that specialize in transporting these types of cars, know how to handle them, and take care of them.
With some company that does moving of any sort (furniture, equipment, etc), you might not get the skills that you want.
But, it really comes down to the car, and how important it is to you. Moving your chevy vega across country? Probably any mover. Moving your 1930 Madame X Cadillac for showing at Meadowbrook? Go with the pros.
My 2¢.
Doc