As a mandolin player, I have discovered that the type of pick (plectrum) one uses makes a noticeable difference to the sound. I personally prefer thick (1mm) plastic picks, like Golden Gate, to get a warmer, "woody" sound, but I have often heard that tortoise shell produces "amazing" tone. I am aware of the ecological reasons for banning that material, and am not looking for one, but I am curious why that material is revered?
The non-uniformity of the density of the material is probably the most noticeable factor in changing the tone. The manufacturing of plastic/nylon based picks creates a uniformly dense product, where as natural products like shell or wood have inconsistencies in the material due to slow growth and building of the product, which in theory will change the response and feedback of the pick.
I have played on reclaimed shell picks (from old hair brushes) and they certainly have a noticeably different texture from plastic picks. For me the tone difference wasn't significantly different from some of the acetal picks that I use.
In some cases there may be some feel or tone difference because the shell picks are hand shaped, not cut from a template, so the leading edge can have a different profile. The shell will also wear differently than a plastic pick.
Dunlop puts out a hand shaped plastic pick to try to replicate this effect. I have used those as well, and they do have a nice feel to them.
Personally, at the rate I use/loose picks I would never have a shell pick for very long any way.