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?
Thickness affects tone. Thickness of strings, too BTW. The smoothness of the pick's edge affects the noticeable string noise. Some players prefer minimal string noise on their attacks for a smooth sound (similar to the well-sanded fingernails of most classical guitarists). Some players like the string noise on their attacks. In this regard it's truly player's choice. If the tortoise shell has a rough or high-friction edge then you'll get that attack sound. I have seen agate picks and other stone used, as well as copper pennies. Use what you like and don't worry about "prized."
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.
A pick made from the shell of a hawksbill has major advantages over a pick made from other materials. Tortoiseshell is less flexible in thinner gauges (speed and tone), naturally lubricated (speed and tone) harder (tone), and more resistant to breakage in thinner gauges. One of shocking developments in synthetic picks is the use of very thick picks in an attempt to get the tone of shell. Because one of the major advantages of shell is rigidity in thinner gauges: Clarence White and Tony Rice. Around 1989 I made a horn pick for Tony Rice. He sent very specific specifications. Exactly this thick. It was nowhere close to 1mm, but probably more than .7mm. (It was also in thousandths!) I can't remember the exact number. It was around 34 years ago. Getting old. But way way thinner than the Grisman pick.