A crest is a component of a heraldic display, consisting of the device borne on top of the helm. Originating in the decorative sculptures worn by knights in tournaments and, to a lesser extent, battles, crests became solely pictorial after the 16th century.
A flag is a piece of fabric with a distinctive design and colors. It is used as a symbol, a signaling device, or for decoration. National flags are patriotic symbols with widely varied interpretations that often include strong military associations because of their original and ongoing use for that purpose.