Why do turntables sound different?

We are not really closer to answering the original question. In essense, the only kind of answer we have so far is they just do sound different. Or, they just color the sound differently and that's why they sound different. The exact science behind it let alone art remain elusive.

The exact science behind it let alone art remain elusive.

I laid out what the engineering principles are. How well a turntable manufacturer adheres to them will determine how neutral their machine is.

We have a Studer A-80 in the studio. Of all the solid state machines we have, it is clearly the best performer (and by that I mean the most neutral).
Tape calibration is not hard if you have an MRL test tape and the manual for the deck. The main difference between consumer decks and pro decks is that the consumer decks are usually set up for flat 1K-10K record and playback, while pro decks are usually set up for flat playback with the lowest distortion in record.

When we recorded Canto General, we used two different machines. Essentially one was tube and the other solid state, both fed the same signal from the mic preamps. That way we were able to audition the difference between the two master tapes. The tube-mastered tape seemed to sound more detailed on both the tube and solid state machines, so we went with that.

Its a simple fact that all tape machines sound different (assuming proper calibration), just as all turntables do (and for that matter, a lot of digital playback).

To defend Ralph against myself, though I only attack him when he is advertising, he is not alone who thinks that modified Technics SP10-MK III with the right arm, cartridge and phono comes very close to the tape sound. I just wonder if top belt drive tables were involved in those comparizons. I think, not sure, that when Fremer was evaluating the Continuum Caliburn he actually said exactly that - closest to tape, and so he got it for himself.
It's easy to choose an open reel deck - just get Studer A80 or 810 or Ampex, and if you cannot any of those - Otari. But with turntables, considering that there are many, it's not easy even if you can audition them. But yes, I would actually compare any table with my deck. In other words, even if you are not going to use the deck, you need it to choose a turntable ! And possibly speakers too. In fact, anything. I don't think many do it this way but maybe more than I would expect.

Anvil, With reference to direct-drive turntables, you wrote, "Coupling a motor to the platter, in my estimation is a compromised design."  You are not the first person in your industry to make such a distorted statement, so I am sensitive to it.  In a direct-drive, the platter IS the rotor of a motor that has its stator mounted symmetrically around the spindle.  There is zero possibility for the coupling to introduce noise, except for the noise of the bearing itself.  Bearing noise is a bugaboo common to all types of turntables. Quiet operation is a virtue of direct-drive, not a problem.  DD introduces other possible issues, but not that one.

As to your statement about DD lacking microdynamics, this can happen if the motor causes counter-rotation of the stator, because of the hi-torque forces involved. (Motor drives platter in clockwise direction but its torque exerts an equal force in the counter-clockwise direction. Per Newton's Third Law of Motion.)  If a tiny amount of counter-rotation of the motor or its parts occurs, this results in a perceived speed error by the servo, and the servo exerts a correction.  Thus there can be a constant hunting for speed on a very micro level that is not readily perceived as pitch error but as a loss of detail or a mechanical sound.  This is why DD turntables do best in massive plinths, IMO.  But also, Richard Krebs has addressed the issue for Technics tables with his internal mods that reduce the capacity of the stator to rotate against its own force.

I don't waste my time arguing for one turntable technology uber alles.  I have found what I like, and I plan to stick with it.  Probably the same is true for others participating in this thread who champion other drive technologies.  

Inna et al, Like any top dogs in any field of endeavor, both the Continuum Caliburn and the Rockport Sirius have their detractors, even among belt-drive aficionados.

I would also think that the belt in the belt drive design should be very important. It's like a cable, in a manner of speaking, that connects two components. 
I can even imagine that at this level my preference might depend on a particular music and record, so I would take both the Continuum and modified Technics, no need to choose, really.