When Aunt Alexandra shows up, she acts (as Scout notes at the end of Chapter 12) as if she had been there "every day of her life." Alexandra quickly establishes herself as an authority ordering Cal to take her bag, then, like an overbearing mother, telling Scout to stop scratching her head.
Scout believes that it was Alexandra's idea to come stay with the children more than it was Atticus's. Alexandra notes that "we" (she and Atticus) felt it was time for the children to have another adult influence, notably a feminine influence for Scout. Scout is quick to think that Cal is the feminine influence in her life but she does not voice this to Alexandra.
Alexandra fits right in with the social structure of Maycomb. She fits right in with the Missionary Society. Unlike Atticus, Alexandra insists upon upholding class distinctions and family history. Alexandra wants to teach the children that they are not run-of-the-mill. In other words, she wants them to learn to behave like their predecessors. However, she instills this lesson with a sense of privilege, essentially telling the children that they belong to a class which is better than some other classes of people in Maycomb. Scout and Jem recognize this kind of elitist thinking is not something Atticus would teach them.
This was not my father. My father never thought these thoughts. My father never spoke so. Aunt Alexandra had put him up to this, somehow.
So, although Atticus agrees that Alexandra should stay, he does eventually tell the children to ignore the importance of family history and class distinctions. This is one of the key differences between Alexandra and Atticus. She thinks certain people are better than others, based on family history and income levels. Atticus is much more thoughtful and treats others with respect; he doesn't form judgments based on class, income, or family history.