I'm a high school english teacher and it is a constant frustration to me the way students have been taught, since kindergarten, that the only difference between a simile and a metaphor is that one uses "like" or "as" and the other doesn't.
A simile is a comparison between two things where one this is said to be 'similair' to the other thing. A metaphor is a comparison where one this is said 'to be' the other thing, or stand for the other thing. The like or as rule was made by lazy teachers and unfortunately co-opted by some reference books. It is true that most of the time a simile will contain like or as, but not always, and simple rote memorization of that rule prevents a deeper understanding of the concepts.
Here's an example I gave my students today: "He looked the way a beaten dog might look."
This is a simile, a comparison between two things, in this case a person and a beaten dog, where they are said to be alike, but not the same thing.
"He was a beaten dog," would be a metaphor.