Typically a softer sponge has a higher ramp at low forces in the deflection/force curve, meaning that it delivers more rebound at lower swing forces. It also generally allows more feel, because the ball sinking into the sponge is more pronounced, especially on slower brushing shots.
The downside is that once the sponge is compressed a certain amount, typically let's say 2/3 to 3/4, it effectively becomes solid and won't add any more rebound force ie: it "bottoms out". So a soft sponge is not as good if you're regularly swinging strongly, because you're going to be saturating the sponge and making it less effective.
Usually a softer sponge is used on the backhand because typical swing mechanics don't allow for as much racket speed and acceleration as with the forehand. Of course players with big backhand swings and great mechanics can be an exception somewhat and may want harder sponges.