One way of thinking about the structure of the Aeneid as a whole is as divided between the first half, in which Aeneas is oriented toward the past, and the second half, in which he is oriented toward the future. The division occurs in Book 6, when, after going to visit his father, Anchises, in the underworld, Aeneas is granted a vision of Rome's future, which fires him up with enthusiasm for the new tasks lying ahead of him. You can also take this same idea and apply it on the meta level. Think about it: Virgil's whole poem is about events long before his time, but it's also supposed to provide direction to the political situation of the day in which he lived. In this way, the poem itself uses the past to shed light on the future.
The war between Turnus and Aeneas represents the conflict between the past and the future.
By killing Turnus out of revenge for a past action, Aeneas fails to live up to the promise of the Roman future as Virgil depicts it.