Say you had 2 (two) balls made from winding rubber bands. Ball 1 weighed 1/2 as much as ball 2 (8 oz and 16 oz. to be exact). The 2 balls are dropped from the same height (10 feet) onto pavement.

Why would the balls bounce back almost identically in height?

I thought the heavier ball would bounce back considerably higher given its initial weight at impact.

Thanks up front for any help you can give.

Use energy conservation.

at the beginning each ball has mgh

at the ground, each ball has \frac{1}{2}mv^2, i.e. \frac{1}{2}v^2 = gh
i.e. the speed, v is identical for both balls and independent of mass.

after collision, if the collision is perfectly elastic (which it is close to in this situation) then kinetic energy is conserved. Obviously it is not perfectly elastic as you hear sound energy, and the elastic bands heat up somewhat. This is the place where any slight discrepancies between the balls will creep in.

Then you do the same thing, all kinetic energy will be converted to potential energy, since h = \frac{v^2}{g}, and both balls have a similar v, h will be the same for both balls.

Only the frictional forces in the middle depend on weight. The other thing that depends on weight (indirectly) are air friction forces, but are negligible for the small distances you are dropping the balls.