How does center of gravity affect the motion of a vehicle

The center of gravity has little affect on the motion of the vehicle, per se. If you turned a corner, the center of gravity would have an effect on how much the vehicle swayed. Other effects on the handling of the vehicle might be seen also.

Ever notice how low to the ground a race car is? Two reasons for this: (1) better aerodynamics, meaning lower wind resistance and improved ground effects (which are not releveant to the question here) and (2) lower center of gravity (CG). Keeping the CG low improves handling - less sway as Perito says - largely by reducing weight transfer to the outer tires during cornering, and conversely reducing the lightening of the load on the inside tires, and also by reducing fore and aft weight transfer during acceleration and braking, respectively. Keeping weight transfer to a minimum while cornering is important as it allows the car to turn more sharply - pull more g's - before the tires exceed their limit of adhesion to the road surface and you start to slide. Hence a car with low CG can accelerate harder through the corners than a car with a high CG. There is also the safety issue - try taking a turn too fast in a high CG vehicle like an SUV and you can flip it - do the same in a low CG design like a sports car and you are more likely to slide it, unless you hit a curb, then all bets are off.