While Atticus concerns himself with the pragmatics of any given situation, including child-rearing, Aunt Alexandra is more concerned with upholding the social customs and values of the time.
For instance, even though Atticus allows Scout to roam freely in overalls or pants, Alexandra considers such clothing and behavior to be "un-ladylike." Alexandra spends a good deal of her time in this novel trying to convert Scout from her tomboyish ways, and attempting to get her to transform into a more feminine girl, clad in dresses and lace rather than denim and leather.
Atticus's attitude toward Scout's personal development is a bit more open-minded, and he allows her to become her own person without the interference of outdated customs. He may be concerned with her manners and her interaction with others, but outside those two areas, Atticus takes a fairly "hands-off" approach to directing his kids.
As is often the case, these siblings differ very much from each other. Whereas Alexandra has automatically absorbed the value system of Southern society without question, Atticus challenges it. One must consider the difference in gender (women were not "supposed" to assert their opinions apart from gardening tips, neighbourhood gossip, and cooking recipes!). Also, Atticus has studied law and has spent a lot of time constructing his values, founded on an intrinsic ideal of justice already very present within him.
Their difference marks their individuality - a brother and a sister subjected to the same environment and influence have simply chosen to be the people they ultimately are. Despite other factors, each is the 'product' of personal choice.