After the incident with the mad dog, Miss Maudie explains to the children: "If your father's anything, he's civilized in his heart. Marksmanship's a gift of God, a talent -- oh, you have to practice to make it perfect, but shootin's different from playing the piano or the like. I think maybe he put his gun down when he realized that God had given him an unfair advantage over most living things."
Atticus, a few chapters later, talks about what true courage is. He is talking about Mrs. Dubose. He says: "I wanted you to see what real courage is, instead of getting the idea that courage is a man with a gun in his hand. it's when you know you're licked before you begin but you begin anyway and you see it through no matter what. You rarely win, but sometimes you do. Mrs. Dubose won.... According to her views, she died beholden to nothing and nobody. She was the bravest person I ever knew."
Not only does Atticus's statement about Mrs. Dubose disabuse the children of any lingering idea that guns are the props of brave men, he also foreshadows his own courageous struggle against injustice in Maycomb, despite being "licked" before he begins.
His choice to never shoot a gun relates to what real courage is, and to being a gentleman who would never take advantage of any living thing.
In a brief summary for Jem and Scout, Atticus implies that guns make men feel bigger than they actually are. So, despite being the best shot in Maycomb County according to Heck Tate and others, Atticus resists the temptation to shoot because (we can infer) that he does not enjoy the false sense of superiority that firearms provide. He would rather remain self-confident on his own terms.
He advises the children not to shoot at mockingbirds, in particular, because they do nothing more than make music for the pleasure of everyone around him. Guns don't really fit Atticus's personality of the intellectual southern gentleman, either -- during this era, they are used mostly by the poorer class of the town, like the Ewells and Cunninghams. Atticus falls outside that social realm as an attorney and as a thinker.
The book doesn't clearly state why, but we can suppose two reasons:
Atticus didn't like to show off in front of other people because he was very discreet by nature.
Atticus was a very kind and gentle man and probably didn't like the idea of killing any animal out of sport.
So in spite of his excellent marksmanship, shooting and killing animals didn't appeal to him; moreover, any form of violence certainly doesn't adhere to Atticus Finch's character profile.