A Noticeable Upgrade from Felt and Rubber Mats
Turntable mats definitely influence sound quality. A superior phono cartridge will easily be affected by the interaction between your record and the surface of your platter.
Common felt mats provide a smooth contact between the record and platter, but they have the disadvantage of accumulating static electricity. That means that felt mats often collect a lot of dust, which can be then transfered to the record. Also the damping ability of felt is very limited, which can be a problem with metal platters.
Thick rubber turntable mats are another solution, but they 'overdamp' the record and have an isolation effect which leaves the static on the record, Audible noise clicks are the result.
"Very Impressive Bass"
A cork mat provides an extremely effective upgrade. Resonances from metal platters are damped more effectively than with a felt mat, cork doesn't attract dust, and they also it also makes solid enough contact with the platter to avoid static.
"In just about every situation where we tried the Cork Mat, it rewarded us with some very impressive bass," notes Richard Black in his TechRadar.com review for July 14, 2009. "Low-frequency instruments seem to gain an extra couple of semitones of clearly audible range, while percussion and special effects have more impact."