A character's dramatic role would be a descriptive label identifying who they are or what they do in the story (no proper names). For example, The Policeman, The Wife, The Estranged Father-in-Law, the Bully, the Youngest Daughter, the Pyromaniac, etc.
A character's dramatic function is defined by the character elements assigned to that character, such as Chaos, Proaction, Trust, Desire, etc. These are the characteristics that can be assigned in the Build Characters window. The dramatic functions indicate the motivations, methodologies, standards of evaluation, and purposes the characters employ in an effort to resolve the story's problem.