Perhaps the reason I didn't notice was that the bit of the Orpheus myth that I always remember the best is Orpheus' journey to the underworld and his attempt to get his wife Eurydice back, which fails when he can't resist the urge to check whether she's following. The movie is interested in this aspect as well, of course, and Christian's journey into the world of Montmartre and the Moulin Rouge is envisaged as a journey into the underworld. However, rather than going there in order to retrieve his lost love, he meets her in the underworld, though his ultimate failure to bring her out from it is more reflective of the myth. There is a sense of the famous look back as Christian walks away from Satine during her final performance, but the look back is not a disastrous act of madness that ruins everything, but a return to sanity that saves their relationship, at least for a last few minutes. In the end, Christian's failure to rescue Satine from the underworld is not his fault, nor anyone else's, but a stroke of fate. I think for me, this is the major departure from the Orpheus myth. The fact that his ultimate failure is his own fault is, for me, the crux of the story of Orpheus in the underworld, so Christian and Satine's doomed romance doesn't quite feel Orphean.
The part of the Orpheus myth that the film does reflect very strongly, though, is the part I always forget about - Orpheus the musician. The son of the Muse Calliope, Orpheus is best known for his musical skill, and of course, this is reflected beautifully in Christian's song-writing skill. Christian's 'enormous talent' is depicted as far ahead of those around him, and his singing and song-writing enchants everyone around him in a very Orphean manner.
I enjoyed Moulin Rouge! for years without realising that it was supposed to be the myth of Orpheus, but I do like the extra dimension it now has with that extra knowledge. The film does very well at depicting the world of Montmartre as a dark, opressive place anyway, but seeing it as Hades does give it that extra creepy dimension. It also makes Satine's struggles to get out into the world above more dramatic and her ultimate failure more heartbreaking. Seeing Christian as Orpheus also gives his extraordinary musical skill an added dimension. Overall, it's perfectly possible to enjoy the movie without knowing anything about the myth, but it's rather fun to watch it with the myth in mind.