Why do turntables sound different?

I am not looking at it from the right/wrong perspective. I simply wanted to initiate a serious discussion. If the mass of the platter is not determined by calculations and listening than it appears to be BS approach. Of course, it's all connected to bearing and drive. That's what I am asking - how the hell is all that bloody interconnected stuff determined? There are only a few elements but I suspect that mathematics and physics behind it should be quite advanced.
And why would Library of Congress and other establishments use mostly Simon Yorke turntables, that do not seem to be especially popular among audiophiles? I didn't hear them use Technics or Walker or that Japanese shining iron you can put four arms on.

I do not know why, but I have heard that it is like this.

Last year I spent a while in a room at the AAA-Forum in Germany
where Conrad Mas from Avid Audio made an interesting test.

There were three of his turntables equipped with the same tonearm-
cartridge combination combined with a 3 input phono stage. The
rest of the sound system was obviously identical.

So many people were in doubt  before the test if there could be more or
less significant differences in the sound of the 3 TTs.
But there were definitively interesting steps going up the ladder in his
turntable portfolio.

Can you expand on steps going up the ladder? I am pondering my final TT. Avid TTs were recommended as a possible option to consider.

I mentally divide up turntables into categories. First we have belt vs idler vs DD. Among belt drives, we have designs that favor massive platter with weak motor (Walker and Notts), massive platter with powerful hi-torque motor, lightweight platter with usually hi-torque motor, or lightweight platter/weak motor. Then you have suspended vs unsuspended, massive plinth vs lightweight plinth. Then you have platter materials to consider. Then you have bearing design and execution, as noted above. Then you have the possibility of magnetic or air levitation of the platter. Then you have direct or indirect application of torque between motor and platter. Then you have platter mats, which can make a huge difference. So, this only covers belt-drive. Which is why it would be most surprising of all if all turntables with pretensions to excellence were to sound the same. Which is why I cannot get my arms around this topic.

@inna 

When I refer to 'mass of the platter', I am referring to suspended mass. That is, loosely speaking, the mass supported by the thrust bearing, including spindle and mat (usually negligible). 

A physicist would normally speak of the moment of inertia, which is the physical analogue of mass in a rotating system, and which is most relevant to rotational stability. Moment of inertia is maximized when all the mass is on the outer ring, which is why some TT have mass distributed on the periphery.

But mass is easier to understand, and is highly correlated with moment of inertia in a solid object. Also, mass determines the weight which must be supported by the thrust bearing. The remaining forces acting on the spindle are radial, arising from asymmetrical rotation and belt tension, and are sometimes dealt with more casually.