Natural rubber is used extensively in higher-end applications (race cars, trucks, aircraft). However, almost all tires are some hybrid of the two. Generally, natural rubber is used in the construction of the tire "carcass" and sidewalls while synthetic rubber (compounded with a gazillion and one other things) is used to construct the tread.
The basic issues are pretty intuitive:
* The supply of natural rubber is constrained by the ability to grow the plants which produce it.
* It's hard to make synthetic rubber with polymer chains as long as those in natural rubber (isoprene), so natural rubber tends to be more pliable and stronger, while synthetic rubber (styrene) tends to sheer off into microparticles.
* However, natural rubber degrades more rapidly when heated and cooled, and is more difficult to control in order to achieve a desired level of stickiness at a given temperature (which is basically what tires are aiming for).
I think that OP's research would be quite interesting to learn about more, as my understanding is that tire manufacturers employ hundreds of chemists who are dedicated full-time to attempting to replicate natural rubber synthetically in an efficient way.