Why do turntables sound different?

I would suggest that quiet bearings would HAVE to be low in friction, period. The greater the friction the higher the noise. Greater tolerances also lead to higher noise levels. The lubricant will aid in reducing friction between two surfaces, but in thicker forms can raise drag in a design (flaw) that may require such to lessen fluctuation noticeability and to lessen the effects of greater tolerances , i.e. , grease. The platter needs to drain, not reflect back to the cartridge.

Your Spacedeck, of the few I have personally heard is more a lively sound with good detail with the cartridges that were installed on them. Heavier mass tables sometimes (not always, try to avoid generalizing as their are always exceptions) don’t play the subtle details as forward that gives the appearance to some as slow. But to the next guy, they sound correct. One thing to remember, ears don’t function the same from person to person. That’s why we have choices of different styles and sound and what music type will also have a factor in the equation of choice as well.

Now, how exactly do the table designers calculate the mass of the platter ? Or they don't but just listen ?

I think your looking at it the wrong way. The quality of the bearing and its ability to carry weight and force quietly and effortlessly, and the drive system to move it will determine platter weight limits. Mass isn't really thought of just in the platter. Mass is usually describing the weight of the plinth required to counter the force a heavier platter spinning creates from that heavy platter. Materials used will determine how it may resonate / and how it will be dealt with. Some suspended tables actually have reasonably thick/heavy platters such as a Michell Orbe SE but many "mass" non suspended tables have very, very heavy platters that require a much heavier foundation as a platform for that spinning mass and may even require a push to start before its own weight takes over.
Cost will always have the biggest determination over choice of materials and design when someone wants to build something and bring it to market. The better materials and the combination of layering/ joining/ machining a quality , balanced platter can be quite costly. As far as the "sound" goes, perhaps you should think of how easy it is to f it all up with a mat placed on it, or how a lesser platter can be tamed with a better choice of mat. Knowing what "sound" you want usually can steer you to the type of table you should be moving toward, not the other way around.

In terms of vibration control, if you look at my turntable I have the following setup:

* The Cartridge Man Isolator (on top of the cartridge)
+ Koetsu RSP
* KAB USA tonearm dampener (silicone based arm dampener)
* HRS Analog Disk (record weight)
= record=
* Boston Audio Mat (graphite record mat)
* Symposium Super Couplers (feet for my turntable)
* Symposium Ultra platform
* Symposium Rollerblock Jr (isolator for the platform)

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.