hyperbole, etc.

Metaphors and similes are very much alike. Both compare dissimilar things as if they were the same. A metaphor presents one thing as if it is the other, but a simile compares dissimilar things using "like" or "as" in the comparison. 

For instance consider this line of poetry by Robert Burns:

My love is like a red, red rose...

The speaker's love (sweetheart) is being compared to a red rose, though when first considered, they may seem to have nothing in common. However, they simply share similar characteristics. His love may have skin as soft as a rose's petal. His love may be passionate, inferred by the use of the color red, which is often associated with passion. The speaker's sweetheart and the rose are different things, but he compares them as if they are the same.

Metaphors can be slightly more difficult to find because the comparison is stated that one thing IS the other. Using Robert Burns' image, consider the following:

My love IS a red, red rose.

In this case the two items are compared but presented as if they are the same thing. 

In both cases, the metaphor and simile are figurative and not to be taken literally. As time goes by a petal will fall off of the rose, but pieces of his sweetheart's body will not fall off, shriveled and dried up. While the rose may have thorns, his love does not. These kinds of literary devices allow the reader to imagine what they cannot see. We will never see his love, but we can assume she is beautiful because she has the characteristics of a rose that we do know something about.

In Edwards' piece, a simile is rather obvious early on.

…my convictions and affections wore off […] and [I] returned like a dog to his vomit, and went on in the ways of sin.

Here Edwards compares his return to sinfulness to be like a dog returning to his vomit. 

For a metaphor, Edwards writes:

The inward ardor of my soul, seemed to be hindered and pent up, and could not freely flame out as it would.

In this statement, Edwards' ardor (zeal, enthusiasm) is compared to a flame. He wants his ardor to grow or reach out like a flame. However, he presents the two (ardor and flame) as if they are the same thing.

One might argue that Edwards' "Personal Narrative" is a metaphor for his walk of faith. He does not employ a great deal of figurative language. Most of what he writes about in his relationship with God is straightforward and no-nonsense. His writing, like the practice of his faith, is uncluttered by things that are insincere. Rather, they are honestly presented just as he would feel he should be before his Maker who knows all things about him.