There has been a lot of "work" put into cone material. Suffice to say there are very little gains to be had in the material of the cone. A heavier cone material results in a lower Fs and a drop in efficiency. However on the plus side, it allows a smaller enclosure and lower frequency extension. Witness the "home audio" driver converted to car audio with the event of "doping" to allow them to run in a ultra violet environment. You may also notice them try to compensate for this with a drop in motor force (Bl) to try to maximise output. (That is from 8 ohms nominal impedance to 4 ohms).
The main problem with cones is resonance within the desired bandwidth, and of cause the harmonics developed over this band. Tests are reasonably conclusive that the choice of cone material contributes very little to dampening. Surprisingly the most important aspect of this is not the cone material but the choice of surround. One of the main differences between "fakes' and the real thing is this termination. If you observe the fake for all intense and purpose it is the 'real" thing. Same cone material, basket size and often a "superior" motor. They normally suck, because they miss the most important part of driver design the "surround termination".
The largest gains on cone material have come from superior manufacturing techniques that have allowed far more consistent results. Expect to see a wide range of materials being promoted to garner your attention, the gains in this area will be small.
We are "spoil-ed" for choice, discern a driver based on the best fit for the application. For me cone material choice is majoring in the minors. The real gains are in harmonic and transient response from a "correct" combination of suspension compliance and surround termination.