A metaphor links a word or phrase with an object or action that's not necessarily related. Example: "to grasp a concept" and "all the world's a stage". A connection between two concepts describes a metaphor.
An analogy, on the other hand, is the relationship between two words. It compares two words and shows how they are related. Example: barbarism:abuse::knoll:mountain. This type of relationship shows a degree. It would be stated as "barbarism is related to abuse in the same way that knoll is related to mountain. When filling out analogy type questions figure out the relationship between the first two pair of words and then try to match the same relationship (degree) to the second pair of words.
Metaphors and analogies are similar in that each is rooted in comparing one thing to another, just as similes are comparisons. The difference between metaphor and analogy is found in their purpose and structure.
The purpose of a metaphor is description and connotation: one entity is described by calling it another. In A Separate Peace, for example, John Knowles describes the tree that plays a central role in the novel by writing that it was a "steely black spike." This is a direct metaphor, saying one thing is another. This metaphor captures the appearance of the tree, but it also creates a forboding tone. Some metaphors are indirect; the comparison is implied.
The purpose of an analogy, however, is not to describe but to explain one idea or entity by comparing it to another and finding corresponding similarities. An analogy is not a simple statement of comparison; it is a comparison that is developed point by point. Analogies are frequently used in instruction. That which is not understood is explained in terms of that which is understood. For example, drawing an analogy (developed comparison) between the human brain and a computer would make the processes of the brain more understandable.