What is the difference between an elastomer and a rubber?

Interesting question!
Originally, “rubber” referred to a naturally derived (organic) material. The
term was first used by an English chemist (Joseph Priestley). Joseph was
working with natural rubber and observed that it could “rub out” a pencil
mark.
In the past, “rubber” was used to describe materials that occurred naturally
in nature; and “elastomer” was used for materials that were produced
synthetically ... silicone elastomers, for example.
An elastomer is a polymer that shows elastic properties. Since the terms
rubber-like and elastomeric mean almost the same thing, the terms “rubber” and
“elastomer” are often used interchangeably today to describe silicone
elastomers.