How to distinguish anology and metaphor?

Briefly, analogy is a perceived likeness between two entities; metaphor is one “figure of speech” which you might use to communicate that likeness.

For example: you may recognize that many Greek and Shakespearean tragedies have a similar structure: a phase of increasing conflict between opposed sides or characters, a major confrontation between the opposed characters, and a phase in which the opposition is worked out and resolved in one character's victory and the other's defeat.

It may then occur to you that this structure is very like the shape of a pyramid isosceles triangle, which rises from a baseline to a central point and then falls back to its baseline. You have then perceived an analogy betweeen a temporal phenomenon and a spatial one.

To communicate this analogy, you may employ metaphors. You name the central confrontation the “climax” —this is the classical name for a figure of speech, which is itself a metaphor: the word means “ladder”. You then name the first phase the “rising action” and the fourth stage the “falling action”.