[RangerScience]

Hmmm good point about the sand deformation but I think you're off on "exactly the same".

AFAICT - The force to bring the ball to rest is exactly the same. As you point out, either the sand or the ball's structure (or the ground's structure, such as a trampoline) absorbs that force. After that however, one ball remains at rest, and the other ball accelerates back up. The force producing that acceleration is exerted (keeping in mind newton's third). If the ball returns to 80% of the original height, then that's 100% of the force to bring it to rest (same in both situations), and then an additional 80% of the force to re-"throw" it, which is only present in the bounce. So in a "thud" there's 100% of the force, and in the "bounce" there's 1xx% of the force.