A simile and a metaphor are both literary devices used to compare one thing to another or to transfer the sense or aspect of one thing onto another. The major difference is the fact that a simile is a type of metaphor whereas not all metaphors are considered similes.
A simile explicitly refers to comparisons that use the words "like" or "as" to connect the two concepts.
Ex. "That house is as big as a mansion" or "He's tall like a tree."
Once more, this can also be considered a metaphor—however every other type does not need to use like or as as part of the connection.
Ex. "Don't cross that bridge before you come to it" is a metaphor for "don't worry about something before you need to."
A simile is a comparison that uses "like" or "as."
"Sally was as red as an apple."
A metaphor is a comparison as well but it doesn't use "like" or "as," and the comparison is more direct.
For example: "Education is a vital instrument of progression in society." Education is being likened to an instrument very directly.
Similies compare two things using like or as. Ex. : She ran as fast as a cheetah.
Metaphors don't use like or as. They give a visualization to imagine. Ex. : You are what you eat.