Both graphic designers and illustrators do design-type work.
Graphic designers will work on design elements and structures, providing a visual message/brand for a company in order to sell a product or service.
Illustrators, on the other hand, will typically do commercial work for companies like comic book houses, publishing houses and advertising agencies. They do a lot more drawing, designing of product packaging, working on book illustrations, creating company logos, and graphic novels.
Graphic design degrees require a concentration in product design, website design, and publication design. Illustration students have some graphic design training, but most of their coursework includes art history, drawing, and painting. The illustrator doesn't have the advanced knowledge of a graphic designer.
If you like to draw and illustrate concepts, illustration would be a good fit. If you prefer to code, make websites, and do detail-type work, then graphic design might suit you better.
A neurologist is a medical doctor who specializes in treating diseases that affect the human nervous system.