It's pretty perceptive to realize that understanding why any ball bounces is the first step to answering the question.
Stretch an elastic band. By stretching it, you are changing its shape. Because it's elastic, it bounces back when you let go.
When a ball hits the ground, the ball changes shape at the point of contact. Because the ball is bouncy (duh, that's what makes it a bouncy ball), the elastic energy in the ball bounces back, pushes back against the floor, and pops up.
Silly putty has aspects of both an elastic object (bounces back) and a plastic object (something that changes shape and stays that way). When it's a ball, the deformation is all right where it touches the floor -- check it, there'll be a flat spot where it hit -- but there's enough elasticity in the nature of Silly Putty for it to bounce back. When the SP is flattened, the contact with the ground changes the shape plastically more than elastically, not leaving enough "spring" for it to bounce.
(I actually teach elastic vs plastic behavior using silly putty in my university classes...)
