Some unsystematic thoughts from an intermediate Latin student:
They too use stock phrases to complete lines and other repetitive devices in their own ways, so I can see how it's possible to train yourself to be able to do it in a craftsman-like way where constructing a metred narrative becomes almost second-nature.
BuiltLikeTaft briefly mentioned Parry and Lord already - their studies of the Serbian guslars revolutionized Homer scholarship in this regard in the 50s. Many features that had previously been decried as qualitatively inferior and even un-Homeric became explainable as deriving from the toolkit of oral poetry.
(1) poetic narrative is (or was) seen as more serious than prose
It was also the target of extreme philosophical suspicion (Plato).
One very simple change is that sentence structure becomes far more complex and the sentences longer, simply to fit prosodic demands.
What language are you talking about? Latin or English? Reshuffle the word order to something more prosa-like and most Latin verse is pretty pedestrian in term of complexity and sentence-length.
They must be useful--or at least fulfilling some purpose--because the epic and similar poetic traditions are so successful in producing great literature.
It also produced incredible amounts of junk (e.g. what little else we have of the epic cycle is far less impressive than Homer), so be careful about survivorship bias.
These are all good and helpful answers, and they make a lot of sense. The only thing I'm not seeing (apart from (3) to an extent) is some objective quality that poetic form imparts to the writing, especially as a result of the formal structure that poetry imposes.
I know you primarily want to think about narrative poetry but to me it would seem easier to start this consideration from didactic poetry (and verse letters). With Lucretius we have at least one author who actively thinks about the difference a poetic form makes in this area (the honey-cup simile) and looking at these forms of poetry might also sharpen one's eyes for the situation and expectations of the readers or listeners (Is an ars composed in hexameters consumed differently from one written in prose?).
Also late antiquity might be a good place to look - with the spread of Christianity Plato's criticism of poetry came back with a vengeance and hardly a Christian poet can do his thing without giving at least some explanation for why he thinks that writing Christian poetry is possible and justified.