'Sound deadener' works in more than one way. The method whitedragon is describing utilizes the viscoelastic properties of the mat (semi-liquid, semi-elastic material) to transform sound waves into heat. Think of it like a shock absorber for sound waves. Since the material is flexible, it will move as sound waves impact it. Since its much heavier than air, the waves will degrade much faster while traveling through the material. That energy from the sound waves is transferred into heat as the molecules in the material rub together, friction between the molecules being the actual shock absorption mechanism. So as you can imagine, sound waves enter the material, some of that energy is transferred to the mat material in the form of heat, the final result being a loss of sound wave energy within the immediate environment. Pretty basic stuff once you get past the whole big scientific sounding word 'viscoelastic'.
The second way sound deadener mat works is by mass loading. The lighter/thinner a material is, the higher the frequency at which it resonates. A thin piece of paper will resonate at a much higher frequency than a thick sheet of lead. As we add mass to a sheet metal panel (or plastic or whatever), we add mass. This lowers the resonating frequency of the object/panel. Since we know that the human ear's most sensitive freq range is midrange, the further we can deviate the resonating freq of a panel from that area, the less audibly noticeable the panel resonating will be. Imagine a piece of sheetmetal hanging by a rope/wire, you hit it with a hammer. It makes a sharp clang/pinging noise. Now you cover that panel with heavy sound deadener, and hit it again. The sound is now more of a dull thud. Even just imagining this made-up scenario, our minds associate a louder noise with the sharper higher frequency clang, and softer noise associated with the dull thud, even though the same energy is being applied to creating those sound waves. The only difference is the frequency, again the thud being further from our ear's "sweet spot", so it is perceived as being quieter. Psychoacoustics at work.
Asphalt, even from back in the old days, claimed to use the viscoelastic transfer method to lower noise levels, but the truth is asphalt mat dries out as it cures, gets hard, and loses what little viscoelastic properties it ever had. Within years, or even months. Butyl rubber is much more viscoelastic than asphalt based mat to begin with, and does not degrade to any noticeable amount even after many years.
I have never seen anyone conduct any sort of scientific to compare the perceived difference in sound dampening between the viscoelastic transfer and the simple mass-loading method. Luckily, butyl rubber mats utilize both efficiently, while asphalt mats rely mostly on mass loading. Between that, and the fact that asphalt mat will eventually dry up, lose even more performance, and eventually fail (long before the butyl mat would), there is no question that butyl mat type dampening pads are the clearly superior product. The only question you have to ask yourself is, is that difference worth the price difference for your application. I wont ever go back to asphalt mat (besides using up the few remaining pieces I have laying around here, for odd jobs), but some people can justify the inferior product based on price.
edit: I should note that I have simplified the viscoelastic effect a bit here, for the sake of the discussion. But its worth noting that its actually the combination of the vicoelstic butyl rubber layer, and the rigid aluminum sheet layer, in a deadening mat that actually function to transfer sound waves into heat efficiently. Neither material would work nearly as efficiently without the other. The two layers rub together, which is where most of the heat comes from as it is transferred from acoustical energy.