Elastic means that I can stretch something and change its shape a great deal without breaking it or tearing it apart.
Bouncy means that something would regain it shape quickly after having its shape changed, thus making it bounce better. It would be elastic, but in a special way.
If something is elastic, it would not necessarily bounce. In fact, if I tried to bounce something that was very elastic, it just might flatten out when it hit a surface and not bounce at all.