In The Odyssey, how are Odysseus' heroism and quests similar to Wart's in The Once and Future King?
This is a very interesting question to think about, because I personally think that there is not that much that is similar between these two characters based on their presentation in these two texts. Of course, both are famed epic heroes whose reputation precedes them. However, it is important to note that while Homer presents Odysseus i the mould of an epic hero, the character of Wart is presented somewhat differently, in a way that seems to undercut his heroic reputation.
The key difference to my mind is the way that The Once and Future King focuses on the genesis and development of King Arthur as an epic hero. Because we see the entire process from start to finish of how the humble and not too bright Wart is transformed into being the legendary King Arthur, the humanity of this character is stressed. This is of course in contrast to the presentation of Odysseus, who starts Homer's epic as a fully-fledged epic hero. In White's classic, the humanity of Wart, as this affectionate but not too complementary nickname suggests, that is stressed.
In the same way, we can equally argue that their quests are different. The biggest challenge that Wart faces is trying to rule justly without the excessive use of power. Wart struggles greatly with this as he tries to learn to preserve peace through not becoming a tyrant and abusing the power that he holds. Odysseus has no such scruples, and uses his power at any stage that pleases him to achieve his goal, which is his safe return and restoration of Ithaca. Both characters are therefore presented as being very different.