A fact is a descriptive statement about the world. We generally back it up with perception and double-check it from other points of view, either by moving ourselves or asking someone else. For example, "I just saw a bald eagle. Did you see it? No? Let me walk over here and see if I can get a better look at it."
A symbol is something which represents something else. It's composed of the sign--the thing itself--and the signified--the thing it points to. So a dollar is a symbol. The piece of paper is the sign. The value it can be traded for is the signified. The symbol "dollar" is both the paper and the value.
The trick here is that the signified doesn't actually exist. It's all in your mind. The word lollipop (as ink on paper or a sound you hear) is a sign for a piece of candy. But the signified is not an actual piece of candy. It's the idea of the piece of candy in your mind. The signified part of a sign is never material but rather psychic.
In a lot of ways facts and symbols have nothing in common. It would be preposterous to confuse one for the other, at least by the definitions. The value of a dollar is not something you can look at, touch, or manipulate physically. It's a collective psychic phenomenon. What makes it confusing, though, is that whether or not someone accepts a symbol is a fact. "Target will trade that dollar for 50 marbles." That's a fact, if it's true. The symbol isn't a fact, but you can make factual statements about a symbol.
But there's no objective foundation for the signified. Target may say the dollar is worth 50 marbles. I might say it's worth 5 marbles, and you might say it's worth 200 marbles. And none of us can be wrong. However, I can be wrong if I say that a dollar is worth 50 marbles to you--it's not, it's worth 200. So I can't get a different perspective to confirm my symbol, but I can test facts about symbols. In this case, I ask you how many marbles you'd give me for a dollar.
The personal acceptance of a symbol is unquestionable by those outside of the mind choosing it. "The color red is symbolic of violence." You can't prove or disprove that statement without the context of for whom red is symbolic of violence. So for example, you could make a statement, "Red is symbolic of violence in Chinese culture" and that statement could be refuted or supported as a fact, because it places it in the context of a specific entity that could accept or reject the symbol.
But what if I said, "Red is symbolic of violence for me." I may be lying to you but there's no way for you to know that unless it conflicts with past claims about symbols.
You can also make factual statements about a symbol's sign: "The color red is light at a wavelength of about 650nn" or "A modern dollar is very hard to counterfeit."
A symbol's signified can also be a mental model of a fact. The word lollipop's signified--the mental image of a piece of candy--probably includes ideas like sweet, sticky, etc. And those material sensations can be confirmed or refuted the next time you pick up a lollipop. The apple falling on Newton's head is a symbol for gravity or perhaps scientific discovery. Gravity and scientific discover can easily be expressed as facts.
So they are two different mental tools which don't have a lot in common, but each is used in the construction of the other.
Of course, the same signs can point to multiple signifieds, and signifieds can in turn be signs of something else. (While signifieds are never material, that isn't to say that signs are always material.) So there can and do exist great chains of signs that branch and web creating our model of the world. But it's a relational model, not a predictive or causal model. Symbols are the components of this relational model in our minds.
We also create predictive or causal models, and those are based on facts. I think the problematic thing arises when we attempt to prove causation through a symbol. Or, alternately, when we attempt to define relationships through facts.
Man, I am more of a structuralist than I ever realized.
So in terms of walrusjester's conversation on U.S. history versus U.S. memory: I have a network of symbols which creates a memory of the U.S. for me, which in turn defines my relationship with that country. In the case of many people, that relationship is a fundamental part of their identity. So the symbols that create the mental model of U.S. memory are deeply important to me, because my identity is important to me.
But what happens that the symbol become so tightly bound to the relationship that in a weird way the sign (Washington chopping the cherry tree) becomes as important or, in the worst cases, more important to me than the signified (honesty in U.S. leaders).
For most of us who find this attitude confusing and frustrating, if a given sign were proven to be non-factual, we would take one of two courses: first we could change the sign into a mythic story, never intended to be based on fact, where the story is told even though it is not true because the message is important. Second we could find another sign for our symbol--perhaps "Honest Abe"--to become the part of our memory that constructs our national identity.
But the argument over the cherry tree, or the nature of the revolution, or the personal merits of the Founding Fathers, is really a miscommunication. The historian is trying to improve the model of causality and facts. The romantic is trying to retain the model of relationships, symbols, and identities. The conflict only arises because the signs of every symbol are asserted to be facts, and the rejection or acceptance of that sign is also seen as the acceptance or rejection of the signified.
On the other side, there may be a problem of carefully studied rejection of the necessity of the relationship--in other words, deliberately goading people by debunking facts that are important to them in consciously hostile ways. That doesn't seem constructive, either.