Metaphors are most frequently compared with similes. The Columbia Encyclopedia (6th edition) explains the difference as:
a simile states that A is like B, a metaphor states that A is B or substitutes B for A.
Where a metaphor asserts the two objects in the comparison are identical on the point of comparison, a simile merely asserts a similarity. For this reason a metaphor is generally considered more forceful than a simile.
The metaphor category also contains these specialised types:
Allegory: An extended metaphor wherein a story illustrates an important attribute of the subject.
Catachresis: A mixed metaphor used by design and accident (a rhetorical fault).
Parable: An extended metaphor narrated as an anecdote illustrating and teaching such as in Aesop's fables, or Jesus' teaching method as told in the Bible.
Pun: Similar to a metaphor, a pun alludes to another term. However the main difference is that a pun is a frivolous allusion between two different things whereas a metaphor is a purposeful allusion between two different things.[6]