Miss Caroline, the sweet first year teacher filled her role exactly. She thought she was book smart. She operated as a teacher how she had been taught to teach, but there's more to teaching than what can be learned in a book, she lacked the human component of understanding and building rapport with her students. She failed to try to understand the school culture, or the culture of Maycomb. Thus, Scout being the most learned in the class tried to interpret Maycomb's ways. This resulted in punishment for Scout as Miss Caroline interpreted her behaviors as trying to show Miss Caroline up.
Atticus, equally book smart, had experience on his side. Although a widower, Atticus was a model parent and teacher because he treated his kids like what he wanted them to be (moral and educated and caring contributors to society), not the babies that Miss Caroline might have thought they were. Atticus taught his children through example. The trial he took on and his ability to put up with the town's insults model this. If you have to pick one that's better, pick Atticus. He produced young people that will make an impact on society for the better.
Scout has an exceedingly hard time on her first day of school because of Miss Fisher. When Scout tries to point out that "Cunninghams don't accept charity..." she is punished, even though her intentions were good. Miss Fisher's lack of cultural understanding accounts for this conflict.
In contrast, Atticus would have informed Scout (in a gentle and reasonable way) not to speak about others' financial condition. Atticus's guidance is usually delivered in an accurate, laid-back fashion, not altogether different from his courtroom persona.
The other difference deals with reading: Atticus has taught Scout to read simply by reading her the newspaper every evening, and Miss Fisher feels that Scout should not know as much as she does about the subject. To defend her opinion, she asserts that Scout's father has been teaching her to read "the wrong way." This also upsets Scout considerably, and in the end, she vows never to go back to school, an opinion she has to be dissuaded from by Atticus himself.
Atticus teaches his children by spending time reading to them. By reading to them nightly he has demonstrated the importance of reading. He passed on a desire that led to his children wanting to read. The book does not say, but I am sure that Atticus would have had to spend considerable time showing his children the words and having them read back to him from early childhood books.
Miss Caroline has just been taught the latest methods to teach reading. She is over zealous as first year teachers often are. She believes what she is saying when she tells Scout that Atticus is not teaching her the correct way to read. I find this funny considering that Scout is already reading.
Miss Caroline and Atticus are both right in regards to teaching methods. Modern studies have proven that children who are introduced to reading early in the home setting on a regular basis respond better and usually become stronger readers. Miss Caroline was right in the fact that for children like her other students with little to no experience in reading, her school learned methods would work well.