So all methods of public transportation you "get on", private transportation you "get in". Or it may simply be because cars were used first in the U.S. (on a widespread level) and the other forms of transportation were more used in England at first. The Americans took the British vernacular when it applied to buses/trains/planes, and the British took the American vernacular when it applied to cars.
I don't think it has to do with public versus private transportation at all. A private citizen can own a own a bus or a plane, and I don't that would change how you would talk about it.