Having a large vocabulary doesn't make you verbally smart. Verbal memory and ability to assign meaning to new words, in addition to the ability to make logical inferences, however does. Nobody is impressed when you pull out your big boy words if you cant properly use them to craft an argument or identify appropriate conclusions from papers.

Having a large knowledge of various proofs doesnt make you smart, but knowing how to derive things from first principles if possible does, or being able to solve new problems using the knowledge of proofs does. Additionally using logic and spatial reasoning in math is also smartness when it comes to being creative in solving geometric problems, or in writing proofs.

Having a good memory doesnt make you smart, but the ability to recall related information and apply it does.

Know the difference between being educated and being smart.