A cultural value is a characteristic or a quality which the people of a specific place (and time) collectively hold as being important to its culture. Your question suggests that there is only one correct answer; however, there are several cultural values which we hold today but are not particularly evident in these two books. While both cultures do value some of the same things, modern American culture values individualism and equality while the ancient Greek culture depicted in The Odyssey by Homer does not.
Though I am no expert in ancient Greek culture, it seems clear that the Greek culture as demonstrated in this epic does not value individualism, one of the core values of modern America. Individualism implies that members of society are seen as unique persons, allowed to choose, for example, who they will worship (if they choose to worship at all) and how they will live their lives. In The Odyssey, everyone must offer their obeisance to the gods or pay the price; even worse, they are at the whim of whatever gods might take an interest in them. In book one, Athena has to plead with her father (Zeus) to allow Odysseus to finally come home.
Did not Odysseus offer you delightful sacrifices on Troy's far-reaching plain beside the ships? Why then, Zeus, are you so angry with him?
Zeus admits that Poseidon is angry with the great hero and will not let him come home. Odysseus has virtually no say in the matter (though we know Odysseus keeps angering Poseidon). However, in this ancient Greek world, man is often not free to choose his own destiny, something every American values and has the right, within the framework of the law, to do. This culture has strict rituals and customs which must be obeyed, not the least of which is showing proper respect and deference to the rather capricious gods; instead of being free-thinking and free-acting individuals, they were constantly trying to appease the gods and do what society dictated they should do.
Another value modern Americans cherish is equality, one of the foundations of this country. In theory, at least, women are equal to men in America; in the ancient Greek society as portrayed in The Odyssey, women are often objectified and certainly beneath the stature of the men. Athena is an extraordinary woman; however, she is not human. These are the females Odysseus mentions in book nine as he recounts his journey:
Calypso, that lovely goddess, tried
to keep me with her in her hollow caves,
longing for me to be her husband,
…in the same way, the cunning witch
Aeaean Circe held me in her home
filled with keen desire I'd marry her,
they never won the heart here in my chest.
These women used their bodies to keep Odysseus prisoner to sate their own lustful desires. At Ismarus, Odysseus and his men killed all the men and “seized all their wives” along with other treasure which they later divided among themselves. When they reach the island of the Cyclops, sea nymphs bring them sustenance (like waitresses or servants). In book one, of course, we have the beleaguered Penelope. While she has been strong enough to endure the suitors for years, they have taken advantage of their superior position as males in this society and have consumed nearly everything Odysseus left her. Eurycleia, one of many female slaves in Odysseus’ home, was purchased for twenty oxen. The women in this story are not generally admirable characters, nor could they be called in any way equal to the males in this epic tale.