Atticus and Alexandra are two very different people. The only similarity I can think of is that they both care about others, but in very different ways. Alexandra cares what others think...Atticus actually cares about the people themselves. Alexandra seems to represent the old traditional South, clinging to her heritage (what little 'fame' she might have), social standing, prejudices, outward appearance, and southern hospitality/graces. She is appalled that Scout acts so much like a little boy rather than being and dressing properly as she feels a lady should.
Atticus on the other hand is progressive. He is considered to be fair, just, and honest. He has new ideas and new philosophies about life, and he raises Scout and Jem in this progressive manner. He allows both Jem and Scout to express themselves and doesn't require that they behave in ways 'worthy' of the Finch name. He is opposed to a false show of etiquette and behavior, and prefers to live openly and honestly with himself, his children, and his neighbors.
In the novel, Scout mentions that this drastic distinction between the two siblings is a result of gender. She says that only a woman could raise them as Aunt Alexandra is trying to, but I think the reason for the difference also has to do with what the author is trying to show in the novel. The differences between Atticus and Alexandra reveal the hypocrisy and prejudices to which some members of society cling.