How are similes and metaphors different

The simile and the metaphor are two literary figures that have in common the fact that they are used to make a comparison.

That is, both compare two words, concepts or terms, but the difference is that:

While the simile links two terms with a comparative nexus (such as: as, similar to, it seems, just like, etc.); the metaphor does not, because in this case one term represents and substitutes the other (terms that are different but share some type of similarity).

For example, the following sentence has a simile:

Here we are making a comparison using the comparative nexus as.

And this one is a metaphor:

Here we are making the same  comparison (the green color of the eyes with the green color of the emeralds). Nevertheless, no comparative nexus is used.