Alexandra considers the Finches better because of their breeding, but Atticus believes who you are is what is important.
Atticus has raised his children to be good people, and to treat everyone with respect regardless of their social class. This lesson is reinforced by how he has them treat Walter Cunningham when he comes to eat, even though he is poor.
Aunt Alexandra is different. She considers “gentle breeding” important. This means that she considers some people more important than others because of their social status. Social class is important to her. She feels that the Finches should act better than others, because they are better than others. She does not approve of the way Atticus is raising his children, and asks him to raise his children her way.
"She asked me to tell you you must try to behave like the little lady and gentleman that you are. She wants to talk to you about the family and what it's meant to Maycomb County through the years … (Ch. 13)
Atticus tells Scout and Jem that Alexandra has asked him to tell his children that they are role models for the community and should ask appropriately. This means, as far as Scout is concerned, that she should act like a lady. She should wear dresses and not pants or overalls. She should also behave in a civilized manner and not an unladylike manner.
Atticus is more interested in having his children treat people, even people like Tom Robinson, in a caring way. Yet he cares what his sister thinks. He needs her help and assurance, and he wants her support. He also wants his children to be well-looked-upon by the community, even if her ideas are not his. So he passes on her ideas, and tries to make sure that both points of view are part of their upbrining.