This is an interesting subject. To begin with, most rubber car mats are not really made of rubber. PVC is the most common material, especially at the low end of the market, while synthetic rubber materials like TPR (thermoplastic rubber) dominate the better mat category. Smack in the middle are rubber mats.
Rubber mats can either be virgin rubber (new rubber), recycled rubber, or a blend of both. Rubber has some distinct advantages. It is extremely flexible at high and low temperatures. This prevents rubber mats from shattering (like PVC mats can) at sub zero temperatures. Also rubber mats, because of their flexibility, tend to lie down better on your vehicle's carpeting. Also rubber is a sustainable material.
The biggest drawback to real rubber mats is they smell. Recently when reading car mat reviews on Amazon.com, I noticed smell was in the top 3 complaints from rubber mat buyers. I can tell you from experience that the smell does not soon dissipate from rubber mats (sometimes it never goes away). Recycled rubber mats often have a stronger smell than new rubber mats.
My preferred 'rubber mat' material is TPR. It combines the temperature and flexibility of real rubber with an often more uniform and superior appearance and no bad smell.
Too bad most rubber mat makers do not tell you what their mats are made of....