Here’s my guess as to what’s going on:
So suppose that you have a rubber mallet and a steel mallet about to impact on a nail, both having the same kinetic energy. They will both contact the mallet and begin to deform, the nail doing work on each mallet as it comes to a rest. However, the rubber mallet will deform more (i.e. a greater distance) than the steel mallet. Since the same amount of work was done on each mallet (they had the same initial and final kinetic energies), since work is distance times force, and since the work done on the steel mallet was done over a shorter distance, we conclude that the steel mallet exerted a greater force on the nail.
Aha, you say, then why did the nail hit by the steel mallet go in further, when the force wasn’t exerted over very much distance? Well, the truth is that the above analysis only really works while we’re trying to get the nail started, and it’s not yet moving. When you’re driving a nail into wood, you’re essentially trying to make the wood grains “fail” in some way. Once you’ve managed to make part of a structure fail (in this case, part of the wood grain), it’s often not too hard to make the rest of the structure fail as well. We all saw a particularly ghastly example of that about five years ago.
So the basic reason you want to exert a steel mallet rather than a rubber one is that it exerts much more force on the nail, and this force is enough to drive the wood into “failure mode” while that exerted by a rubber mallet isn’t.
Hope this helps…