First of all, there are the names. Orfeo is simply the name Orpheus adapted to Middle English. Incidentally, in Greek, the name would have been pronounced "or-FOOS"; the trisyllabic Orfeo is obviously by way of the Latin pronunciation "OR-fee-us." Similarly, Eurydice's name has been Englished: the four-syllable name with the accent on the third syllable and the long eta on the end of it became a three-syllable name in English. In Middle English, final "e" was usually pronounced, but was already beginning to acquire exceptions, as here; meanwhile, the accent shifts to the first syllable (one can tell by the meter that Heurodis is pronounced "HYUR-uh-diss"). Sir Orfeo is lord of Traciens (given by Tolkien as Tracience. This is the Middle English version of "Thrace." In order to make the story "at home" in English, the poet explicitly equates Tracience with Winchester.
In the ancient story, Eurydice was walking with her companions when she was bitten by a viper and died; her spirit descends to Hades. In the medieval poem, Heurodis is taking a nap with her companions when she is shaken by a nightmare in which the king of Faerie says he will come for her. Sir Orfeo and his knights surround Dame Heurodis, but she vanishes from their midst. Note that she does not die; she is magicked away to a different sort of Underworld. Later, Sir Orfeo sees people in that underground realm who have died, and others who have been thought to have died but maybe haven't. There is always some ambiguity in old English thought over whether the mortals resident in Faerie are dead or merely enchanted.
Orpheus heads directly for Hades, as if he knows right where to find it. Sir Orfeo leaves his kingdom to the rule of a steward and goes questing after Heurodis. This allows for a lot of interesting quest material that appealed to the English. Some of this was later picked up by Tolkien himself in The Hobbit, as when Sir Orfeo catches glimpses of the king of Faerie out hunting:
There often by him would he see,
when noon was hot on leaf and tree,
the king of Faërie with his rout
came hunting in the woods about
with blowing far and crying dim,
and barking hounds that were with him;
yet never a beast they took nor slew,
and where they went he never knew.
Eventually, Sir Orfeo sees Heurodis as part of the trooping fairies and he follows them into the underground kingdom. There, he plays for the king of Faerie as Orpheus played for Pluto, and the result is the same, though in the classical story, Pluto is moved to pity, while in the poem the king of Faerie is held to his pledged word to reward the unknown harper with any boon he can ask. (The tears running down Pluto's cheeks is also an image appropriated by Tolkien, when Luthien sang for Mandos the Doomsman of the Valar, and moved even him to pity.)
Finally, in the classical story, Orpheus lost Eurydice, by turning around and reaching for her too soon. She was not yet free of the realm of the dead, and she vanished from him, leaving him to live out the rest of his life alone. Sir Orfeo and Dame Heurodis live out their lives in love and lordship, "happily ever after," as the saying goes, though they have no children. Orfeo's faithful steward reigns after him at Winchester.
What I find interesting is that the Middle English poet doesn't just retell the story in its classical setting, though he could have. English poets before and since have told tales that way, including the story of Troy. But instead, the poet sets the classical story firmly in the legendary England of his day's consciousness. And thus we deal with fairies, not gods; with magic, not miracle. This changes the story from one of the ineluctibility of death to one of human heroism. The classical story could not end happily and be a story told by the ancient Greeks and Romans; the medieval poem could not end with the fairies winning and sustain the interest of the English.