In the Aeneid, the themes of Duty, Religion, and Family are very closely intertwined. The nexus of all these ideas comes in the epithet "pius," by which Virgil regularly refers to Aeneas. Although it's related to our word "pious," this Latin word also includes a strong sense of "devotion to one's family." So, when Aeneas is on his mission to Italy, he is performing a service for his gods, for his ancestors, for his descendents, and for the other Trojans under his command. Some of the Aeneid's main drama arises from conflicts between Aeneas's sense of duty and her personal desires – as when he temporarily falls under the spell of Dido. In the end, though, duty wins out – though Virgil doesn't shy away from depicting how painful this can be.
The main difference between Aeneas and Turnus lies in how much each cares about duty.
Aeneas is less motivated by duty than by a sense of responsibility; he doesn't act as he does because he has to, but because he wants to.