You are correct, although most Hard Rubber used has at least some synthetic (plastic) in the mix.
Hard Rubber is usually used today for intermediate or better horns as an alternative (and a VERY suitable one in my opinion) to wood.
Plastic/resin is used almost solely for student instruments (although I have played several of the better plastic bodied models, and rate them better than the more "traditional" clarinet crowd typically does).
The performer can often detect some differences in the way a clarinet sounds based on material, but very few listeners can, based solely on that parameter. The same is largely true with mouthpiece material.
I hand make various bagpipes (one of the most conservative, tradition driven music fraternities), and that community has experimented with just about every wood and and other material for the components (involving both conical and cylindrical bores, single and double reeds) for quite some time now. Many makers now offer alternative materials to natural wood, usually Delrin/Polypenco. The difference in tone is minimal to non existent.
African Blackwood/Grenadilla, Honduran Rosewood, and Cocobolo (with Lignum Vitae and Ebony being popular in the past) are the most popular woods for these pipes as well.