Why do turntables sound different?

Look at Michael Fremer's set-up in that small room with the Continuum TT and big Wilson speakers! I bet he's not too worried about vibrational interference, since the Continuum is a hefty beast indeed!

Actually, Mr. Fremer wisely gets quite nervous about vibrational interference. That's why he uses the super duper Minus K negative stiffness isolation stand for the Continuum. The new Obsidian version of the Continuum incorporates a lot of vibration isolation in its design, including magnetic levitation for the platter and a dedicated isolation system for the tonearm. The Continuum might be a hefty beast but the Earth is a heftier beast. 

'Taking care' of vibration will have some effect on resolution, which may or may not make the turntable sound better. Given the limitations of analog playback, it's highly subjective and system-dependent.

The phono system of your hi-fi is just that, a system unto itself.  The turntable, its physical relationship to and with the room, the arm, the cartridge consisting of a stylus and electrical generator, and the load on the cartridge together make up a unique and one-of-kind electrodynamic system.  Change any element of that system and you will change, sometimes only in very subtle ways, the final aggregated performance of such a system.

Unfortunately each element of the system is also a bit different than supposedly equivalent or same element.  Replace any element in your phono system with an identical element, say the arm with another arm of the same make, model, and age, and you will be changing the dynamics of the reproduction system.  You may or may not hear the difference, but it is there.

Since the chain of events goes from the room on one end, and the cartridge loading on the other with whole bunch of subtle mechanical links in between, this is a gestalt where the whole is greater than just the sum of the parts.

There is no formula for audio GESTALT, so science is not going to jump on that band wagon.

Mankind science is a finite observation of infinity.  We can only hope infinity in the long run, will have its day in court.

So for this finite point in time, using our ears, is the best final judge for our own gestalt systems.