Atticus by the generic word daddy when she says toJem she is going to brag about Atticus's shooting skills: " [It] Ain’t everybody’s daddy [who is] the deadest shot in Maycomb County."

However, when addressing Atticus directly and most often when talking about him, Scout and Jem use his name, Atticus. Though the reasoning for this is not explained, there is a dignity to his name that does not adhere to the term daddy, which, though a term for father used across the United States, sounds like dialect in the context of Maycomb. Atticus wants his children to treat him with respect—he has also taught them to address him as "sir"—and his name has a suitable gravitas.

Using Atticus helps Lee set her exemplary central figure apart from the other daddies in Maycomb. This functions as a rhetorical (persuasive) device: while Scout is Atticus's adoring daughter, her estimation of him is based in the reality of how Atticus Finch really behaves as a person, not a little girl's daddy worship.

In To Kill a Mockingbird, Scout and Jem call their father by his first name, Atticus. This appears rather odd to many readers, who cannot imagine calling their own parents by their first names, for it seems disrespectful or distant. In this case, however, the exact opposite is probably true. Scout and Jem refer to their father as “Atticus” out of a sense of great respect and intimacy.

Notice that the children call most adults Miss or Mr. and then their given name, Miss Maudie or Mr. Arthur, for instance. This is a sign of respect and deference. The children recognize their place as children and the adults' place as grown ups with this language. The use of the adult's first name is common while the addition of the Miss or Mr. adds a sense of propriety.

That sense of respect carries over into Scout and Jem's address of Atticus. They are not equal to their father, and they know it well. Their father, in fact, has significant authority over his children. Yet he rarely exercises it. He prefers to teach them through open discussion and questions and answers. He wants them to obey but also to understand why they must obey. Scout and Jem respect him greatly for this, for they recognize that he makes them think critically. By calling him “Atticus,” they combine respect with this recognition that they are called to grow in maturity and knowledge.

Further, the children's use of their father's first name suggests intimacy. They don't call him “Mr. Atticus” as they would if they were referring to another adult. They drop the “Mr.” out of intimacy. Scout and Jem clearly love their father very much. He is not just any “dad” or “papa” to them. They love him for who he is as a person, for they know him well thanks to his encouragement of deep discussions even about difficult subjects. They can come to him with any issue, and he will treat it with respect and seriousness. He also deals with each of his children as an individual with his or her own personality and needs, and they, in turn, recognize their father for the unique individual he is.

It's not reasonable to think that Jem and Scout called Atticus by his given name because of an equality issue. In the South, at the time this amazing novel is set, equality wasn't possible between child and parent any more than equality was possible between black and white. That is the answer in my opinion. Atticus defended Tom Robinson, not because he thought there was a slim possibility he would exact justice on behalf of Tom, but because it was the right thing to do. It wasn't popular. It wasn't endorsed by the current culture, but Atticus couldn't do otherwise. Instinctively, his children knew this about him, and although the current climate forbade them to call him his given name, they did, because, quite simply Atticus was too real, too large, too important, to be called anything other than he was.

This writing device was a symbolic way of honoring what was right, real, raw, and true, despite prejudice and social norm. It parallels the trial in the novel and Harper Lee displayed genius.

Scout does call Atticus "my father" when she is recounting her childhood experiences from an adult perspective.  However, as a child, both she and her brother indeed did call him Atticus. 

To understand how radical this was, one must understand a bit a bit about Southern culture, where, to this day, many children would never dream of calling their parents by their first names, and often that Miss, Mrs, or Mr, is followed by "ma'am" or "sir."

There are several theories about why Scout and Jem were permitted this familiarity.  One suggestion is that this lack of manners is indictive of Atticus' poor parenting.  The children grow up without a mother, raised largely by the maid Calpurnia instead.   Atticus is so busy with his law practice and distracted that he never bothers to teach the children proper social rules.

Another take is related to Atticus' own beliefs in the power of children and the unearned respect adults are often given.  The children call Atticus by his first name in order to cultivate a more open, rather than authoritarian, relationship.  Atticus wants to keep the pathways of communication open between himself and his children.  Allowing them to call him by name is a way he keeps their relationship more level.