Tow rope is specifically made to be elastic because it allows for diffusion of energy when the rope is tightened.
Imagine towing with a chain, which has essentially no elasticity. While the chain is loose, the car in front can move freely forward while the car in back is stationary. At the instant the chain tightens, the car in back would go from zero to the speed of the car in front instantly, causing a very uncomfortable jerk, and putting an immense amount of force on the parts attached to the chain.
Its impossible to keep a chain perfectly taut the entire time while towing so at the very least this would make for an uncomfortable ride and at worst do serious damage to both vehicles.
An elastic rope in contrast stretches when under tension and allows the rear vehicle to slowly accelerate to the speed of the front vehicle.
A tow bar is different because it is rigid and therefor always taut.
This is the same reason why rock climbers use elastic (or dynamic) rope rather than static rope for climbing. When falling on a dynamic rope, the climber slowly decreases speed from a freefall to stopped. With a static rope, the climber would stop immediately the instant the rope becomes taut, exerting a large amount of force on the climbers body, and the bolts holding the rope to the wall.
Edit: In answer to the second part of your question, it looks like the rope you linked to in the image was not made for towing. If you buy one made specifically to tow cars there will be less stretch so you won't be bouncing around so much