In literature, what is the difference between symbolism and allusion?

In literature, “symbolism” and “allusion” resemble each
other in some ways but differ from each other in other ways.  Distinguishing between
them is made all the more difficult since “symbolism” can be (and has been) defined in
so many different ways.  However, here are some similarities and differences that may be
relevant to your question:

• Symbolism and
  allusion both communicate meaning more subtly and indirectly that outright statement. 
  For example, a lion can symbolize courage without a writer needing to say: “This lion
  represents courage.” However, if a writer presented a cowardly lion, he or she might
  possibly be alluding to the similar lion in The Wizard of Oz,
  especially if the writer’s lion says

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Courage. What makes a king out of a slave?
Courage.

What makes the flag on the mast to wave?
Courage.

In other words, if a
later text clearly echoes an earlier text (or, in this case, film), allusion is
involved.

• As has just been suggested, allusion
  depends on the reader’s knowledge of some earlier specific text, whereas symbolism does
  not. If I write a novel in which the main characters are named Adam and Eve and they are
  forced to leave an attractive place, I am certainly inviting readers to assume that I am
  alluding to the Biblical story of Adam and Eve.  However, if I present an attractive,
  innocent young man and young woman who are forced to leave an attractive place because
  of a misdeed they commit, I may be alluding to the Bible, or I may
  just be using the general symbolism of a kind of paradise lost – an archetypal story. 
  Using clear echoes from an earlier text would involve allusion. For instance, if I asked
  about the young couple, “who first persuaded them to revolt in such a foul and
  unattractive way?” someone who had a good knowledge and memory of John Milton’s
  Paradise Lost would recognize that I am alluding to a particular
  line from the opening book of that poem.


• Recognizing and
  responding to symbolism requires no great knowledge or education on the reader’s part.
  If I write a poem in which the sun rises, birds sing, a gentle breeze blows, and flowers
  begin to unfold, it takes no great education to assume that I am symbolizing such ideas
  as life, renewal, rebirth, and happiness.  However, if I write a poem featuring a story
  about an old carpenter named John, a clever and lecherous student named Nicholas, and an
  attractive young wife named Alisoun, an educated reader will assume that I am alluding
  in some way to Chaucer’s “The Miller’s Tale.”


• Symbolism,
  therefore, often transcends cultures (light symbolizes knowledge in many cultures);
  allusion tends to be more culturally specific.  Thus, if I begin an essay by writing
  “Fourscore and several years ago,” an American reader who is reasonably well educated
  will know that I am alluding to the opening words of Abraham Lincoln’s “Gettysburg
  Address” (“fourscore and seven years ago”). A reader from China might not recognize the
  allusion unless an editor provided a footnote calling attention to it and explaining
  it.

Therefore, symbolism tends to be somewhat
universal, whereas allusions tend to be rooted in knowledge of specific earlier
texts.