I read and hear metaphor used with a variety of meanings. A metaphor is a figure of speech applied to something to which it's not literally applicable. Worse, the two words get tossed around as if they mean exactly the same thing.
A simile is a metaphor, but not all metaphors are similes. That is, metaphor is a broader term, and a literary device that transfers the qualities or aspects from one object to another. Example: I tried to write but my mind was an empty screen.
A simile is a comparison but uses like or as. In the above sentence, the metaphor becomes a simile this way: I tried to write but my mind was like an empty screen. My favorite simile is from Robert Burns who penned these words, "My love is like a red, red rose."
I recently read this metaphor by playwright Lillian Hellman: "I cannot and will not cut my conscience to fit this year's fashions." (Notice her correct usage of will in the first person showing determination.)
In short, both words compare two things of different classes. Some grammarians insist that a simile makes the comparison explicit or concrete. Perhaps they're correct, but I find it difficult to see the distinction.
One more thing is what we call the mixed metaphor. I'm surprised that writers aren't aware of using them. This happens when we combine two or more incomparable figures.
Example: We try to sweep thorny problems under the rug but they continue to bob up.
Here's a better way of saying it: We try to weed out thorny problems that thrive no matter what we do. Thorny problems, weed, and thrive keep the metaphor clear.