Every time someone asks this, I think of another question: Can you hear the shape of a drum?
Is there actually any serious advantage to any of the more advanced materials for mid/bass drivers over simple doped paper? And for HF drivers, is there really much of a (dis)advantage for various metal designs or ribbons vs doped silk/fabric?
Besides the costs of the material and the required tooling?
-Most of these materials are dense and heavy, more so than paper and other "soft" materials in similar configurations. They are not made thin enough to retain reasonable efficiency for mid-high audio frequencies. Beryllium has a low density with high stiffness to weight, but this only makes for 1-1.5 dB of difference compared to aluminum.
-High material stiffness, such as corundum or diamond, is a result of the tight and uniformly bonded lattice of the material lacking internal friction. The stiffness and density tend to overwhelm the small losses intrinsic to the material. The defects in the structure of certain softer materials, such as magnesium, weaken the material but increase internal losses.
-Forms with great geometric stiffness constructed of materials with high modulus/weight ratio have less bandwidth than softer cones and reduced power response and less suited for wideband driver duty. Seen a different way, the flexure of the cone offsets the natural increase in directivity as wavelength decreases relative to radiator diameter.
Ribbon tweeters are made out of ordinary materials driven by the usual Lorentz force. The magnetic circuit arrangement lowers motor force BL²/Re compared to typical high-end Neo tweeters, but the mass is small for the surface area, keeping efficiency reasonably high. An advantage is the more uniform excitation of the membrane, suppressing resonant modes that might be obvious if locally driven by a former or coil. Usually, the available excursion is small, resulting in a disappointing non-linear response at the lower range of the driver and high cutoff frequencies.
The contribution of the membrane material and geometry are mainly in the linear parameters: the axial response and directivity. As for nonlinear parameters, the main contributors are motor force variations and inductance effects. These come from the loudspeaker motor; the membrane choice is less critical to nonlinearity.
Most loudspeakers operate each drive unit in a very limited frequency range, so the question of sound quality has more to do with the integration of the components into a whole unit.