These two literary figures of speech are not necessarily related, though they can interchangeably be connected. An allusion is a direct or implied reference to some sort of person, place or event; it can also refer to a literary (or artistic) passage or work. The placement of such an allusion within a written passage is usually just a simple or casual mention, leaving the reader to determine the reason or connection presented by the author. If the passage is detailed in depth, it is usually defined as a reference, rather than an allusion. An example of an allusion would be Harper Lee's mention of "Those Bellingraths" by the avid gardener, Miss Maudie, in To Kill a Mockingbird. (Bellingrath Gardens is a famed botanical site outside Mobile, Alabama.)
A symbol, meanwhile, is the representation of an object, concept or idea that is meant to represent underlying meanings, or something aside from its literal meaning. For example, the color black is often used to symbolize evil, while the color white is regularly used to symbolize innocence or purity.