In Lee's To Kill a Mockingbird, why do Atticus and his children get such different reactions from Mrs. Dubose?

Atticus understands the generation from which Mrs. Dubose has come, and he also understands her underlying problem, whereas the children do not. Cleary, Mrs. Dubose does not approve of some of the things the children do and say, as well as their appearances, especially Scout's.
When the children pass by her house, Mrs. Dubose often sits in her wheelchair on her porch. Scout may say, "Hey, Mrs. Dubose." However, Mrs. Dubose does not return the greeting. Instead, she criticizes both her looks and her speech:

"Don't you say hey to me, you ugly girl! You say good afternoon, Mrs. Dubose!" (Ch.11)

When Jem refered to their father as Atticus, "her reaction was apoplectic." She considers Jem and Scout as "the sassiest, most disrespectful mutts who ever passed her way." And, like many "cranky, old people," Mrs. Dubose is suspicious of what Jem and Scout are doing. One Saturday she asks them what they are doing, all the while thinking the worst of them, and believing that the children are impudent.
But, when Atticus walks with the children and they approach Mrs. Dubose's house, Atticus takes his hat off with a grand gesture, and he waves "gallantly."

"Good evening, Mrs. Dubose! You look like a picture this evening." (Ch.11)

Then he replaces his hat and extends wishes that she will have a good day on the following day. It is at this point that Scout believes her father to be the "bravest man on earth."