Being an object of culture, referenced in all media and all different levels of artistic expression, just about everyone knows something about the Odyssey. We have a cultural familiarity whether we've read the Odyssey, read excerpts of it in school, or just encountered it in references in movies and comic books. We all have an idea of what the Odyssey is about and what happens in it. Well, we're wrong.
The Odyssey is a story of the strife endured by Odysseus as he struggled against the gods to return home to Ithaca from the Trojan War. However, of the ten years it takes him to get back to Ithaca from Troy, seven are spent in the strife and struggle of being the lover of the nymph Calypso, where she fed him ambrosia, had sex with him every night (and not just regular sex, but nymph sex), kept him from aging, and promised to make him immortal if only he stayed with her. (Did I mention it was nymph sex?) That's 70% of his time abroad. May all of your struggles and strife be 70% nymph sex.
While we're on the topic of what Odysseus actually spent his time doing during his trip, he spent a whole lot of narrative time with the Phaenicians, who were not cannibals, or opium addicts, or man-eating monsters. No sea beasts. No deceitful women. No angry gods raining petty vengeance upon a powerless mortal. Rather, they are a wealthy and generous sea merchants who treat Odysseus to a massive feast, imply that he could marry the daughter of the king, and shower him with more gifts than he won in all of his plunder of Troy (which of course, went down with his ship) before returning him to Ithaca on a ship so fast and so smooth that he sleeps through the entire journey.
Furthermore, a full third of the poem takes place after Odysseus has returned to Ithaca and involves his plot to deal with the suitors, a goal “nudged” along by Athena. To recap; 70% of the journey was spent as the boy toy of a nymph, another big chunk of the story takes place on Ithaca, another substantial percent is taken up with the Phaenicians, and there's a whole bunch of stuff about Telemachus that doesn't even involve Odysseus at all. The big events from the Odyssey that we all know about even if we haven't read it; the Cyclops, the Sirens, the Lotus Eaters, Odysseus in Hades, Circe and the whole turning the crew into pigs thing, are a very small part of the actual story as it is told.
Cultural memory is a process of extrusion; as an artifact whether it's an epic poem, story, person, or idea is transferred around a culture its complexity is shaved off and members of the culture are aware of a simplified version of the entity. What a cultural artifact becomes when most in the culture do not have direct contact with it, is important, not just in understanding Homer or Shakespeare, but in more general issues of society. How many of our political decisions are based on the cultural artifacts of the American Revolution? How many of our politicians appeal to concepts of “freedom” and “liberty” without any examination of the documents and events that built those concepts? The most direct example of this is the constant claim that the United States is a “Christian Nation,” a claim that can be made when the distinction between, “a spirituality based in Christian mythology,” (which is what most of the Founders actually had) and “Christianity” is lost or ignored.
I haven't met anybody with the time to read every major readable cultural artifact and the point is not that everyone should feel obligated to read everything (there's a lot of Shakespeare out there, and frankly the Federalist Papers get a little dry after a while), but that we realize that what we know of all of these artifacts are simplified versions. This is fine when enjoying art or entertainment that references these entities, but this is not fine when, say, making a policy decision or establishing a personal belief structure. In terms of important decisions, it seems reasonable to ask people to do a little research.
Oh, and the Trojan Horse thing; doesn't happen in either of them. A bard mentions it in a song about the Trojan War (a song Odysseus requested). Nor does Achilles die his famous death. The Illiad actually ends with the funeral for Hector, the only honest to god decent human being in the entire story (No, seriously, Agammemon is an arrogant jerk, Achilles sulks in his tent while his friends die because he didn't get the slave girl he wanted, and well, Odysseus himself sacked an innocent city on his way home from the war and executed servant girls who were born after he left for Troy for sleeping with suitors.) who valiantly fought hordes of invaders and spent his last night before he knew he was going to die with his wife and infant son. And what does he get for his bravery and general decency; he gets his dead body dragged back and forth in front of the gates of Troy behind a chariot. In the extruded versions of these epic poems it's Achilles and Odysseus who are thought of as the heroes, but after actually reading them, Hector is the one worthy of respect.