Symbolism is the broad category and allegory is under it in hierarchy. Anything in literature can be a symbol, and any symbol can have a wide range of meanings and interpretations. It's a one to many relationship.
An allegory is more specific and limited. An allegory only has a one-to-one symbolic function. In Animal Farm, the pig represents a specific person (a certain Communist leader), not an archetypal representation of a an authoritarian figure. To me, allegory is an inferior art form because its symbolic scope and range are so limited.