I think "on the left/right" or "to the left/right" or "to your left/right" are most common for giving directions or making observations. In this case, there does not have to be both a left side and right side. I think using "-side" implies that there is *both* a right-hand side and a "left-hand side" of something, whether that thing is a road, a building, a painting, an equation, etc. For a car, you could say, "the left-hand side of the car" or "driver's side" and "right-hand side of the car" or "pasenger's side." (Of course this is for a car that has the steering wheel on the left side. Use the opposite if the steering wheel is on the right side, or right-hand side.)

No there is no difference, they are the same.