Compare and Contrast Virgil's Aeneid to Homer's Odyssey in regard to the story of the return from the Trojan War.
The motivations for the respective voyages are different. Aeneas, a Trojan warrior, is one of the few to escape his home city while it burns at the hands of the Achaeans. He now has no homeland, but as a noble warrior with a divine mission, he must establish a new one. That homeland will be the city of Rome.
Odysseus, on the other hand, is one of the Achaeans' victorious warriors. Having tasted victory, he can now return home. He doesn't need to establish a new homeland; he already has one: the island of Ithaca, where he is king. Odysseus' quest is more simple, then, but his subsequent journey is anything but.
Aeneas and Odysseus undergo journeys that demand much of them; they are trials of both physical strength and courage. Things are made more difficult by the attitudes of certain gods. Odysseus incites the wrath of the sea-god Poseidon by blinding the cyclops Polyphemus, his son. Aeneas, for his part, is hated by Juno, who knows that he is fated to destroy her beloved Carthage. During the epic siege of Troy, each god and goddess had their own personal favorites, and that doesn't change in the Odyssey and Aeneid.
The gods' favoritism causes both warriors problems at sea. Poseidon constantly intervenes to make Odysseus's journey considerably more difficult. He can't kill Odysseus, but he can certainly make him suffer, and that's exactly what he does. It's Poseidon who's largely responsible for keeping Odysseus from reaching Ithaca. In punishing him, the vengeful sea-god is also punishing his family.
Juno, for her part, is forever meddling in Aeneas's mission. She unleashes a terrifying storm against Aeneas and his fleet, forcing them to make landfall at Carthage. Aeneas's subsequent love for Dido diverts him from his mission, eventually leading Jupiter and Venus to send Mercury down to remind Aeneas of his duty.
Try as they might, the gods cannot hope to thwart the respective journeys. Both Aeneas and Odysseus have divine missions to fulfil which nothing can prevent them from accomplishing.
The first thing to keep in mind is that the two epics are written from two opposing points of view: the victorious Greeks (for Homer) and the defeated Trojans (Virgil) who are destined to become the Romans. One thing the two poems have in common is the epoch making nature of the war; in both epics, the various battles and heroes are known by everyone from the Sirens to Dido in Carthage.
The various stories in The Odyssey involve the heroes trying to return to Greece. Some, like Nestor, are favored with a swift passage home. Some, like Ajax, are destroyed due their hubris. And others, like Menelaus and Odysseus, have long, purgative journeys that take years to complete.
The Aeneid is different. Aeneas and his crew are among the few survivors of the destroyed Troy. He is given a mission to rebuild a new city in the west, but what he finds out later is that the new city is not to be a new Troy. Rather, Aeneas has to give up the idea of being a Trojan and get on with the business of establishing something new--the Roman Empire. So, although Aeneas and Odysseus both sail west and although they encounter some of the same adventures (such as the Cyclops or going to the underworld), the purpose is different.
Much more could be said about this interesting topic, but I hope this, along with a review of the plots of the two epics, is a start for you.