Both make comparisons, but a metaphor compares one thing to another straight up, while a simile uses "like" or "as."
The word metaphor comes from the Greek metaphora "to transfer." With a metaphor, an idea is transferred from one word to another. It's implicit, like in this metaphor from Flannery O'Connor, "He had measured five feet four inches of pure gamecock." But don't mix them — mixed metaphors get confusing. Don't put all of your eggs in one doghouse. Wha? Here are some examples of the word itself:
Driving is such a metaphor, literally and figuratively, for freedom — as it is in the movie — so I'm amazed when some people never learn it. (Los Angeles Times)
Never mind what Lemmy said — with respect, 'Ace of Spades' can be viewed as a metaphor. (BBC)
Then it was all about finding the right analogy or metaphor for the way to tell an audience. (New York Times)
A simile is similar but it always uses "like" or "as." In fact, the word simile comes from the Latin for "a like thing." A simile's comparison is explicit. Just like that old joke from Fat Albert, "You're like school on Saturday: no class!" But seriously folks, here are some examples of the word in action:
Clouds roasted like marshmallows; everything — eventually — scorched beyond simile. (The New Yorker)
It's like McDonald's,' she said, pleased to have landed on a simile that an American reporter would surely appreciate. ( New York Times)
A metaphor is direct — Rudolpho is a cow! But a simile can soften the blow — Rudolpho is like a cow. Use them in descriptive writing or any time you're feeling sassy.