An allegory is a story type in which the characters represent archetypes and the story line itself can be interpreted as a representation of something else. So in this respect an allegory is one form of symbolism, or a sub-category of it. The thing being represented can be universal or specific. An example of the first would be Bunyan's Pilgrim's Progress and for the second, Orwell's Animal Farm.
There are several theories as to the interpretation of Hawthorne's "The Birthmark." On a universal level, it represents the impossible quest of absolute perfection since imperfection is the very stamp of humanity. On a more culturally specific level, one theorist sees it as a sexual complex of the male in terms of female bodily functions, including menstruation. This would be a taboo subject at the time, and symbolic inference would be about the only way to approach the matter.
Check out the bibliography list below, along with the other eNotes references, for more insight into this aspect.