Rather than bouncing, it hits the floor and flattens. All of the organized motion of the falling ball becomes the random motion of jiggling molecules. The random motion of jiggling molecules is a measure of thermal energy. The putty gets warmer, but it doesn't bounce. Now suppose you drop a rubber ball.
Rubber is made from long-chain polymer molecules.
When you hold the ball in your hand, these long molecules are tangled together like a ball of molecular spaghetti. Atomic motions within the rubber molecules then return them toward their original, tangled shape. Much of the energy of the ball's downward motion becomes upward motion as the ball returns to its original shape and bounces into the air. The energy in the ball that isn't converted into motion becomes warmth. Rubber balls are elastic because they return to their original shape.