If an arm is not stiff enough and/or it does not have enough self damping (does not effectivelyt dissipate the energy of its vibration as heat), the vibrations imparted by the cartridge will feed back excessively to the cartridge and this will affect the sound. That is why, designers try hard to maximize stiffness and internal damping within the constraints of having to keep the arm from being too massive.
But, theory aside, not everyone likes the sound of those arms that maximize both stiffness and damping. I have not heard it, but, a friend heard the SAT tonearm (exraordinarily stiff and damped) and thought it sounded a bit too clinical and lifeless. Some tonearm manufacturers offer an assortment of different materials used in the tonearm shaft to tune the particular resonance of the arm to a particular cartridge or personal preference of the owner (e.g., Shroeder).
If you have ever played with liquid damping of tonearms (changing the level of fluid in a damping trough on an SME or the level in the bearing cup of an arm like the Basis Vector), it is quite easy to hear how changing the amount of damping affects the sound. It is hardly the case that maximizing damping results in the best sound.
I know a lot of people would like improvement in audio to be a simple case of increase this or decrease that, but, it is actually far more complex than that. So much of it involves "tuning" and matching things in a complementary fashion and making the right compromises.
