There are the obvious contrasts of gender (Jem is a boy, Scout a girl) and age (Jem is the elder, Scout the younger). But the more important differences include personality, gender stereoptyping and maturity.
Jem is calmer and more self-controlled, whereas Scout is fiery and blunt. One might cite the incident on the first day of school when Scout attacks Walter because he "makes her look bad" in front of the teacher. She then insults Walter, who is a guest in her home, for the way he chooses to eat.
Jem is typical boy of the time, rolling tires, playing ball and impatient of his little sister's presence. Scout is a tom boy and not at all lady like. She wears overalls instead of dresses, gets into fights and spits out the first words that come into her head with little regard as to their effect. Her favorite playmate is Jem, who is both male and her brother.
Lastly, Jem is the more secure of the two, perhaps because he is older and therefore more mature. He does not seem to exhibit any fear of the changes brought about by the new school year. Past experience has perhaps taught him that his relationship with his father is secure. In contrast, Scout is concerned about losing her close relationship with Atticus, as is demonstrated by her reluctance to continue going to school since her teacher has instructed her to tell her father "not to teach her anymore."