The fundamental difference lies in how each views the residents of Maycomb. Atticus views each family, like the Cunninghams, as an essential part of Maycomb. Remember, early in the text he helps Mr. Cunningham with his "entailment." And Mr. Cunningham pays him as best he can - not with money but with fire wood and nuts. Central to Atticus's beliefs is the idea of crawling inside someone's skin and seeing the world from their point of view (he often tells Scout to do this). Because he has the ability to see the world through the eyes of Mr. Cunningham, Atticus can find something in him to respect (for example, Mr. Cunningham will not accept charity - nor will any of his children). Atticus wouldn't classify the Cunninghams as trash. They make the best of what they have. Notice how he makes Walter feel at home early on in the book when Scout has him over for lunch. Walter and Atticus talk about farming and life. Atticus doesn't judge him nor does he look down at Walter.
Alexandra, on the other hand, lacks the ability to view people from any perspective other than that of being a Finch (which means she is of better stock than most residents of Maycomb). This is why she tells Scout that she can like the Cunninghams all she wants, but they are trash and they not to be treated as friends or invited over. She is much more interested in the social perception of her family than Atticus is. But also remember that Alexandra would never have taken the Tom Robinson either.