What are the major heroic accomplishments of Achilles and Aeneas, and how do they differ?

Achilles was the hero of Homer's Iliad. The entire epic is focused on the "wrath of Achilles", or the period when Achilles withdrew from the fighting because he was angry that Agamemnon took his war prize. What made Achilles a hero was his prowess as a fighter. He is portrayed as brave and powerful, with a strong, if somewhat narrow and selfish, sense of honour. His abilities as a fighter made him able to turn the course of a battle by sheer physical skill and strength.

Aeneas, the protagonist of Virgil's Aeneid, is a more complex and modern character. As a prince of Troy he is responsible for leading his defeated people to safety. He is constantly torn between his duties as a leader, the commands of the gods, and his own desires. While Achilles is a flat character, typical of oral traditional epic, Aeneas is a more rounded character, with a rich internal life, who develops over the course of the poem.

How does Aeneas in the Aeneid differ from Achilles?

Aeneas is a character who is defined by piety, and indeed, is referred to as "pious Aeneas" at various points in the text. In both the Aeneid and the Iliad, characters are often defined by one dominating character trait. Odysseus is defined by his cunning and wily attitude, for example, just as Achilles is defined by his inability to master his emotions and his anger and petty characteristics when annoyed. In contrast to Achilles, who in many ways is presented as an overgrown child who is given supernatural strength, Aeneas is defined first and foremost by respect towards the gods and his betters. One of the most important examples of this comes when he refuses to leave his father behind when he flees burning Troy in Book II:

Did you suppose, my father,
That I could tear myself away and leave you?
Unthinkable; how could a father say it?
Now if it pleases the powers about that nothing
Stand of this great city; if your heart
Is set on adding your own death and ours
To that of Troy, the door’s wide open for it.

This piety is both characterised by the duty he owes to his father, but it also demonstrates the way that, in this instance, Aeneas actually reminds his father of the responsibility that he has as a patriarch to go on living and continue to be the leader that so may look up to. Aeneas thus repeatedly shows himself as a character who knows his place in the order of things, both in the human world and the divine world, and he shows himself to be subservient to both the gods and those who are in authority over him. This is in marked contrast to Achilles, who crashes through life determined to act in the way that he wants to. Even though he is fighting for Agammemnon, he withdraws in a sulk when he claims Briesis, and is only tempted back to fight when Patroclus is killed. Then he savagely kills Hector and desecreates Hector's body. Aeneas is a character defined by the control he has over his emotions, whereas Achilles is the opposite.