A hammer delivers an impulse (i.e., a change in momentum, calculated as the integral of force over time, https://en.wikipedia.org/wiki/Impulse_%28physics%29) to its target. In the case of a hard, rigid hammer striking a hard, rigid surface, the impulse will be a relatively large force delivered for a relatively short duration.
A mallet (including the dead-blow mallet, described in your question) is designed to deliver the same impulse as a lesser force over a longer period of time.
The difference lies in where the energy is going to be absorbed when you strike a piece of wood---say, part of a piece of fine furniture that you are building. The collision is going to be mostly inelastic in either case.
I don't know the physics---I'm not a physicist---but my gut reaction is that the extreme force delivered by the hammer will cause a lot of stress in the wood right at the point of contact, and that the wooden surface is likely to be permanently deformed. Part of the hammer's kinetic energy therefore will be absorbed in the damage site.
I'm guessing that with the mallet, the lesser force and the longer time interval mean that more of the hammer's energy will go into accelerating the whole piece of wood and, ultimately will be absorbed by friction in the joint that you are trying to pound closed.
Sorry, but I don't know how to model the situation with mathematics. I'm thinking that there would be stress and strain tensors---way over my head.