They both mature, which is to be expected when a novel occurs over a three year period of children's lives. They also both grow more respectful of all people.
Scout, in particular, began the novel happy to tell Miss Caroline what was what, and to beat up any boy who made her mad. By the end of the book, she learned to think before she spoke at the Missionary tea, and to let things go, like when the boys went swimming without her.
Jem, used to just get quiet in the beginning of the book when he was upset, but by the end (for example when the kids discuss the types of folks) he more quickly articulates his thoughts and ideas.
To Kill a Mockingbird stands among many novels in a genre often give the name of Bildungsroman, or the novel of maturation. In such a novel, the central character(s) are taken from an ingenuous state and brought to an experienced and enlightened state, resulting from a series of misadventures which compose the narration. In the three years with which the novel is concerned, Scout comes to abandon her childish superstitions of "haints" and spectres such as Boo Radley and Mrs. Dubose; and, she learns to accept people for who they are as individuals and not according to the gossips or stereotypes. She also learns about religious and racial prejudice, as well as learning much about her own father, whom she at one time has viewed "as a feeble old man."
Both Scout and Jem learn about virtues, also. From Mrs. Dubose, Atticus, and Boo Radley, she apprehends the real meaning of courage. For instance, after Mrs. Dubose withdraws from morphine and dies with nothing for her pain, Atticus tells Jem who has been reading to her until shortly before her death,
I wanted you to see what real courage is, instead of getting the idea that courage is a man with a gun in his hand.
The children learn about fortitude from their father, as well, when he shoots the rabid dog, and when he takes on the case for Tom Robinson, saying that he could not face his children if he does not do so. When the mob comes to the jail, Atticus does not waiver in his defense of Tom's right to a trial.
Aware from his father that justice should be administered regardless of race and class, the idealistic Jem undergoes a maturation after the trial of Tom Robinson as he is faced with the incongruity of what is right with reality. Scout also learns about hypocrisy when the Missionary Society feigns concern for the natives in Africa while they criticize their maids. On his part, Jem learns of hypocrisy, too. He also learns to act maturely. For instance, when Jem tells Atticus that Dill is in their house, Scout becomes angry with him for "telling on Dill," but Jem maturely realizes that Dill's mother wil be worried when she discovers that he is missing.
Both Scout and Jem, as well as their friend, Dill, learn several moral lessons in To Kill a Mockingbird. As she stands on the Radley porch, after having learned to "consider things from his point of view," Scout even concludes that there is little else for her and Jem to learn--except, perhaps, algebra.