When we first meet Dill, he proudly proclaims, "I'm Charles Baker Harris ... I can read." Immediately, this marks him out as a young, childish, rather loveable character. The pride with which he tells Scout and Jem that he can read is indeed a very childlike trait, as is the very simple language with which he speaks.
In the same scene, Jem shows how mature he is when he notices that Dill doesn't want to talk about his absent father. When Scout tries to push the issue, Jem notices that Dill blushes and is uncomfortable, and he tells his sister to "hush." In this scene, Jem is thus presented not as an equal to Dill, but as more of an older-brother figure. He is the responsible, sensitive, mature one.
Later in the story, Dill and Jem discuss Boo Radley and the possibility of drawing him out of the house. The conversation they have is a good illustration of how the childishness of one contrasts with the increasing maturity of the other. Dill tries to shame Jem into going into the Radleys' front yard and tells him, "You're scared." Jem responds by saying, "Ain't scared, just respectful."
Dill then repeats the accusation that Jem is scared, and Jem says "he reckoned he wasn't [as] he'd passed the Radley Place every school day of his life." The calm, reasonable replies that Jem gives to Dill here are a good example of his increasing maturity. He does not respond to Dill's childish provocations impetuously or irrationally, as a younger child might. Instead, he gives reasoned, sensible replies, again emphasizing his role as the responsible and mature one, in contrast to Dill's role as the childish, impulsive one.