An allegory is a story told by a symbol's POV, or a highly symbollic story. An example is the character named Mercy in Bunyan's Pilgrim's Progress. Not surprisingly, that character acts in ways which exemplify the symbolic name.
Symbol is less obvious, but is another means of representing an idea through another means. Gatsby's flashing green light has been described as symbollic of his flashy lifestyle and of his affluence...yet it can be described as symbolic of Gatsby's envy as well (as in the phrase "green with envy").
To contrast examples, Gatsby's symbolic green light adds subtle clues to our understanding of Gatsby's character; but, Bunyan's character, Mercy, is symbollically named (since mercy is not a name but does have obvious meaning) and acts out her part in the allegorical novel accordingly. We can discuss and disagree with the symbolism of Gatsby's green light, whereas we would have nothing left to say about the allegorical character, Mercy, whose name says it all.
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.
The difference between symbolism and allegory is rather thin. On the one hand, an allegory would have to be a symbolic text, on the other, symbolism is a necessary technique of an allegory.
Both symbolism and allegory relate to a process of representation where one thing stands for another like light representing life and darkness death, to use a cliche.
A text like Waste Land by T.S. Eliot is an allegory of the War-ridden Europe though it hardly refers to the scene directly. Symbolism is a basic technique in the text whereby the themes like alienation and absurdity are addressed for example in the London Bridge section in the first part of the poem.
Symbolism is generally the literary technique or trope or device that facilitates the allegorical layer of a text while allegory is rather the generic model or the genre of the text.
If Coleridge's Kubla Khan is an allegory of the poetic process, then symbolism is the device that has helped set up the allegory and sustain its individual symbols that make up the allegory overall.
An allegory is a simple story to help one understand the deeper meaning of a teaching. Something symbolic has characteristics and qualities of the symbolism. For example the story of "The Boy Who Cried Wolf" is allegorical of someone who regularly tries to gain attention for no real reason and who will not be believed when it really matters (deeper meaning). Whereas "A wolf in sheeps covering" is symbolic of those who outwardly look innocent and mild but they are really hateful and spiteful (characteristics and qualities).
Symbolism and an allegory are basically the same thing the only difference is that an allegory is more specific. Symbolism is a very broad topic ; it can be a character representation something or just representation; there are different types of symbolism.