The distinct but interrelated concepts of "signs" and "symbols" are important in a variety of academic fields, including linguistics, philosophy, psychology and mathematics. Unfortunately, the words can mean different things in different areas of study and to different authors. In mathematics, for instance, "symbol" may refer to a variable, while a "sign" denotes whether a number is positive or negative. For the most part, however, a "sign" is some unit of communication that stands for something else, while a symbol is a unit of communication imbued with deeper and more complex meaning.
A sign is simply something that stands for something else. For example, the word "cactus" directly correlates with the idea of a spiny desert plant, so the word is a sign signifying the plant. Signs are not limited to a single meaning. The sign "cactus" might refer to a specific prickly pear plant or to an entire species. The word still correlates directly, or stands for, something else.
Symbols are best understood as signs that have deeper and more complex meaning. Signs translate directly to objects or ideas, but symbols have something of a life of their own. The Christian cross or the Star of David, for example, do not have a direct translation. Rather, they carry symbolic meaning that can differ in different contexts. Worn on a chain around the neck, the cross might be a symbol that a person is Christian. To the person wearing it, however, the cross might be a reminder of personal faith or a means of identifying with a particular sect. This deeper meaning distinguishes the cross as a symbol instead of a sign.