Hyperbole is a special kind of exaggeration.
The difference between hyperbole and plain old exaggeration is this: When people are exaggerating, they might believe what they’re saying, and even if they don’t, they probably want YOU to believe it. But when people use hyperbole, they know they’re doing it, and they expect you to know it, too.
Let’s say I want to tell you about a banana spider that once fell from a high ceiling and landed on the book I was reading. Legs included, the spider was about as big as my fist.*
Exaggeration: That spider was the size of a grapefruit.
Hyperbole: That spider was the size of a military helicopter.
See the difference?
Hyperbole should not be confused with a hyperbola, which is not similar at all.
*That sounds like an exaggeration, but it isn’t. Sweet dreams.