Symbolism in your story will give it another dimension. Not all readers will catch it, but those who do will appreciate it. Much of the great classics in literature are heavy on symbolism. I remember being in my Senior Lit class in high school studying such things and wondering if the author had intended that when he or she wrote it. I have written things that I looked at later and realized they were symbolic while at the time that was not intentional. At any rate, if you want to write an award winning story, it should probably have it at some level.
Using weather to reflect or enhance a story’s mood is a classic, albeit unoriginal, way that actually isn’t true symbolism, at least the kind I’m talking about. It doesn’t have to be an object, though it can be. Often it is the plot itself, for example illustrating that not all prisons have bars, but can be a person’s mind. Science fiction is often used in this manner. If you’ve ever read “Alice in Wonderland” and didn’t catch the symbolism (perhaps because you were too young), you might want to read it again. Like nursery rhymes, literature is often used as a form of thinly veiled political protest.