On the answer
Much of the structure of this answer is based on the very clear history of The Odyssey and The Iliad written by Nicolas Bertrand in a 2009 Article (PDF). The primary sources discussed are the following, and I will try to reference them more closely in successive edits, but this is a process that takes time, as I relied on translations to write the original answer):
- Epic Singers and Oral Tradition, Lord, A,B, Cornell University Press, 1991
- The Making of the Homeric Verse: the collected papers of Milman Parry, Parry, M, Clarendon Press, 1971
- Die typischen Scenen bei Homer, Arend, Weidmann, 1933
- Homeric Questions, Nagy, University of Texas Press, 1996
What's a text? This is a question that needs to be answered to be able to understand this question fully. What you're asking is basically, why we don't we have a diff /homer.txt /oral.txt? There is nothing like a signed version of The Odyssey. Whatever version you read was probably compiled (I'm not saying translated, I mean compiled) in the 19th century.
This implies that if we want to compare the version you're reading with oral tradition, we need to identify which version you're reading - a problem one seldom has with texts from post-medieval era, and has to a much latter degree with texts that post-date the diffusion of codices by Ancient Rome.
Come on, I want an answer
At the birth of the Odyssey (and the Iliad) is a long-lasting oral tradition. This sophisticated tradition has few current-day equivalent in Europe, but luckily enough, in the early 20th century, Perry and Lord, who were as curious as the OP but had no access to the Internet, went on asking living bards for their input on what the Oral Odyssey (this is my term, not theirs) could have been: epske pjesme, the oral epic tradition in Yugoslavia, was still lively and they had many interviews with practicing bards that confirmed their intuition (intuition shared with a third guy called Arend, working separately, who apparently preferred libraries to the Balkans).
What they got from their discussion confirmed two great intuitions those guys had:
The Odyssey and The Iliad were not written to be... written. They were written down after being composed to be learned by heart. This was first identified by identifying then studying structures called "Homeric verses": those are rhyme structures repeated with similar semantic content (generally a character and an associated adjective). The canonical example of this is "Swift-footed Achilles". One would have three or four such structures for each character, that he could rely on when coming back to the character. They have an evident mnemonic function, which tells you a lot about how close to Oral tradition the version we can read is, but they are also a great "writing" tool: they are prepared to fit in Homeric verses well, allowing to keep the flow whenever the story takes a new twist. Those structures are actually very old and are sometimes shared with Sanskrit (making it ridiculous to argue that we're talking about western culture here)!
Even if you go away from the language itself, the oral character of the story is evident: all boat landings, all fights, share a common structure so that poets only have to remember the scheme once, and decline it according to what's important in a given example. Of course, seeing this from only a single version of the story (whichever "Homer" version you found) can be difficult, but it's confirmed by the study of the above mentioned epske pjesme. This kind of system speaks very much against The Odyssey being written by a single man. It also says a lot about the stability you may expect between versions: they will all share the same structure, and big discrepancies would therefore be about omitting or detailing a given event in the story (small discrepancies between oral tradition and a written text not being very likely to survive time...)
So come on, where are those discrepancies?
The same studies on Yugoslav poets that "proved" the existence of a mnemonic framework throughout The Odyssey and The Iliad were studied long enough that we know how little they vary: several years apart, poems built around similar verse and story structure were recorded.
This allows for poet to compose quickly and efficiently improvisations on a given structure. What "modern" users of the same technique tell us is that this was used to give out acts, various and varied performance of a given theme, with both great consistency and great diversity.
Then how come we have only one version?
Now that we know for sure that those were oral texts, how did we get down to the single version we read today?
First, it's a transcription. The best way to see the Odyssey or the Iliad is as a movie. A movie is not a script, it's not even the sum of all scenes of the script that were played out once. It's a collage of chosen bits that were recorded and distributed. Of course the base material (the actual scenes recorded) is important, but you can't go back to them from the written (recorded) piece. Nor is it realistic to think we recorded a single performance of a poet: both "books" are far too long for that.
Once we agree The Odyssey is just a movie, we need to know when it was shot.
Shooting The Odyssey
The question is: who put The Odyssey on paper?
First option: The Scriptist. There was a predating cultural substrate, but the guy who wrote done the two poems also composed them. The main argument for this is in my opinion weak, ie that poems of such wisdom and complexity could not have been conceived without the help of writing
Second option: The Oralist. The poems were dictated (by one or more poets). The tenants of this option (among which Lord and Powell) witnessed that the texts that were dictated to them by the Yugoslav poets were much better when dictated than when recorded, as it forced the poets to pace themselves and extend the sentences. We can use the analogue of theater actors knowing they are recorded for a movie, asking to shoot a scene again or including bonuses that are not in their usual performance.
Retaining this option gives for a very affirmative answer to your question: the two poems are not different per se from the oral tradition, but they consist a more organised and exhaustive version of the poetic wisdom that brought them to life.
Third hypothesis: The texts transitioned slowly from oral to written to written and unified. This is Nagy's hypothesis. He gives a very precise timeline (which in my opinion is quite irrelevant here) but the basic idea is to have the text put down in three phases. At first, as described above (just a writing framework and tool), then more and more "standard" versions would circulate. In a third time, some of these performances would be recorded in a written form. In a fourth time, those written down would be aggregated and standardized before, ultimately, a "fixed" version appears (for The Odyssey, we can date this to Aristarchus's edition in 150 B.C.)
I think this last option is preferable: we should be able to imagine an oral tradition that is slowly merged into a nascent, written tradition, before for both political and cultural versions a central version emerged, that would then vary only slightly - "corruptions" and "cleaning" periods, basically through spread (for corruptions) and study of many versions to make a clean one (for cleaning or bottlenecks). The text you know comes at one of those bottlenecks, and its incredibly long history as a "single" cultural object explain no variations of it have gained wide tract.
Notable exceptions to this are details of the Trojan Horse that survived in other texts/the popular culture, and a lot of mythology that we know in contradictory/more detailed version, because both these elements had a cultural significance to those over time that transmitted the text to you, from Athens scholars to Roman and Egyptians to Arabics and Byzantines, to Italian scribes of the 15th century, to French scholars of the 19th and British scholars of the 20th...
You know of one version because every effort has been made to present a coherent, written version of a text that was initially probably a collection of oral performances.
