The word "sign" can refer to a number of different things, but when contrasted with "symbol" it is defined by Dictionary.com as "any object, action, event, pattern, etc., that conveys a meaning." In the same vein, the same source also defines a sign as "a notice, bearing a name, direction, warning, or advertisement, that is displayed or posted for public view: a traffic sign; a store sign." In these senses, signs point to something or instruct someone about something. For instance, a "Stop" sign or a street sign designate a required action or a place. Signs are, thus, physical things.
In contrast to a physical sign, symbols represent something and are, thus, not physical. While a sign designates a place or an action --- telling someone what to do or informing them about what something literally is --- a symbol is more abstract. According to Dictionary.com, a symbol is "a word, phrase, image, or the like having a complex of associated meanings and perceived as having inherent value separable from that which is symbolized." For example, the well-known image of a dove symbolizes peace, rather than just depicting a dove itself.
Sometimes "sign" and "symbol" can be used as synonyms. In these cases, the thing referred to is something that "stands in" for something else --- they are both used abstractly. For instance, a handshake could be referred to either as "sign of peace" or a "symbol of peace." In both cases the word does not refer to the action literally, but rather abstractly. The key in using these words synonymously is to ensure that either one refers to something in a non-concrete way. It may be confusing if a writer uses the word "sign" to refer to both a physical object --- for instance, a stop sign --- and a symbol. In such a case, it is best to use "sign" to refer to a physical object and "symbol" to refer to an abstract representation.
Common symbols that are not considered signs include a dove to represent peace; a cross to represent Christianity or, in some cases, medical organizations; or even punctuation. Some common signs include "Stop" signs and other street signs; wall signs in public places such as bus stations, malls, and airports; and signs denoting notable spaces such as cities or countries.