I'm a Dummy, Tell Me About Turntable Mats

You can always remove adhesive from the materials used for platters. Lacquer thinner will take it right off. The way the record is clamped to the turntable is more important than the mat. The mat should be firm, have a depression for the label and be 113/4 " in diameter, just under that of the record so that the records lip hangs over. There are two decent ways to clamp the record down. The first is reflex clamping. A thin washer is placed over the spindle and the camp only contacts the record at the rim of the label. This flexes the record into the mat. The best example is the Sota reflex clamp. SME and Kuzma tables come with reflex clamps. The second is vacuum clamping which requires a turntable design for it. With reflex clamping you do not need a rim or ring clamp. Ring clamps do not work well and are bulky and difficult to use. I have seen records and turntables damaged by them. They are also additional mass on the bearing. They will not work with any end of record auto lift that I know of.

@atmasphere, all that is very true. However, it the record is not firmly coupled to the mat it does not make much difference. The weight of the record is not enough aside from the fact that few records are perfectly flat.

I just ran a little experiment for fun. The stylus noise is easily twice as loud with the vacuum defeated as against the vacuum on! It is just barely audible with the vacuum on and my ear right next to the record. This is with a record of standard thickness.

The stylus noise test not only picks up noise generated by a vibrating record but also from a vibrating tonearm.

@lewm, it is not an echo. It sounds like a very high frequency copy of the music. What is vibrating can not emit anything from maybe 6 kHz down. I'm just guessing. With the vacuum engaged I can just barely make out cymbals. With the record loose it is 2X louder and I can make out a little more. It is still very quiet. You have to put your ear next the rim of the record to hear it. I remember my old Zenith which I got when I was 4 years old. I use to marvel how I could hear the music with the volume turned down. It was pretty loud. You could hear it across the room. 
Remember the old Victrolas? They had a sowing needle in a shank fixed by a grub screw. The shank was attached to a diaphragm in the throat of a horn. Same thing but no diaphragm or horn. You are hearing whatever is vibrating. The less you hear the better is the control over distributed resonance from the stylus and cantilever vibrating. The cavity around the stylus can act as a horn which is why many manufacturers stick the cantilever out in thin air. That resonance is reflected back at the cantilever to be heard. You really only want to listen to the record. Nothing else.

The best record playing set ups hardly make any noise at all. There may be a few out there that make no noise at all. I have never heard one make absolutely no noise but I have heard a few get very close. 
Just don't laugh when you see a guy with a tonearm stuck in his ear. 

The point of all this is, the job of a turntable mat is to help control resonance. It is to help keep the record from vibrating under the stylus.
Because records are not flat a mat can not do it alone. People figured that out and started using record weights, then record clamps, then taking a lesson from record lathes, vacuum clamping. I am comfortable with saying that vacuum clamping is overall the best. I do not know if it sounds better than reflex clamping but it does results in the best control over the record with the least effort and stress on the record. 

@lewm, It is not really record noise. You are hearing the mechanical system of the record/stylus/cantilever/tonearm vibrating. Ideally, all the energy spent in moving the stylus should be translated into an electrical signal. Some is turned into heat and some is turned into noise. The noise is a problem because it reflects back on the structures creating it which you theoretically can hear amplified along with the music. 
Let us call this "tracking noise." Tracking noise has been mentioned by others as a measure of the quality of the playback system. Certainly, those old ceramic needles were very loud and I remember my fathers old Rec O Cut with its old Empire cartridge was easily audible 3-4 feet away.
My current combination is very quiet but my main point is that clamping the record properly makes an obvious improvement. 
I had a Transcriptors turntable at one point decades ago, the one that set the record on 6 brass weights. I hated it. It was pitiful in comparison to an LP12. I blamed the platter for that.
I'm sure Chakster and Rauliruegas can comment on this.

Thank you atmasphere, I think you said it better than I. 
Lew, I know of no hard data that says it is a definitive sonic problem. We both know it exists because we hear it directly. If we hear it the assumption would be the stylus does also. I apologize if I contradicted myself. But it is useful to define our terms so we all know what we are talking about. Record noise is what you hear in the background behind the music including dirt and scratches. Let us say that tracking noise is the noise generated directly by the stylus tracking the record which in all respects is just like that diaphragm in the Victrola. The mat has to dampen the resonance without reflecting it back. IMHO the record has to be held down tightly as the mat is not doing it's job if the record is partially in the air. I think the fact that disabling the vacuum increases tracking noise is telling. You use the Sota reflex clamp which performs the same function as long as you are using that washer over the spindle. Keep your amp off and listen to the tracking noise with and without the clamp. See if you get the same results I did. 

@fsonicsmith, most people would think this is all mishegas, even those of the not so Jewish persuasion. 

Just why do you think mats change the sound? Could it be because they dampen the record in differing ways? If so would they be dampening some sort of resonance? Where would you guess that is coming from?

Turn off your amp and play a record. Put your ear down next to the rim. What do you hear? Is sound vibration? What is the job of a phonograph cartridge? Is it not a vibration detection device?

Sei Gesund,
Mike