Alexandra and Atticus both adhere to rather rigid moral codes. The main differences between the moral view of each character stems from what they view as the crux of good and bad behavior.
Each bases moral judgment on a different, basic criteria.
Atticus...bases his opinions of people on their behavior and not their background. Unlike Alexandra, who calls poor people like the Cunninghams "trash" because of their social station, Atticus tells his children that any white man who takes advantage of a black man's ignorance is "trash."
Where Atticus is concerned with fairness and justice, Aunt Alexandra is concerned with status. She has reasons for viewing the world as she does, but these are reasons that Atticus (and Scout) can find no agreement with.
Alexandra...is a conservative woman concerned with social and class distinctions and bound to the traditions of the South.
Alexandra's concern with family distinction and with status is at odds with Atticus' humanism, a trait that by-and-large defines his character.
For Atticus, morality is defined by principle and action, not on appearance. This is why he is willing to sacrifice his stature in the eyes of his community to conduct a thorough defense of Tom Robinson in court.
The principles of "right behavior" are bound to notions of fairness for Atticus. He gives up shooting so as not to enjoy an unfair advantage resulting from his gift. He strives not to use his position in his own favor. He aims for compromise with his children. In all these examples, we see a dedication to fairness as the basis for moral behavior and justice.