Having finished both the manga and anime last week (I do plan on rewatching the English dub versions of episodes 8-12), I can say for certain that the reason why the anime turned up different was most likely Studio TNK wanted to have the story have more mecha stuff and do things Kaishaku weren't doing (giving the Orochi Necks not named Souma, Tsubasa, and Chikane backstories) .

Now, on the differing endings, I imagine that while Studio TNK may have devised the anime version's ending to allow Kaishaku to come with something different for their manga's ending, both do have the implication that Ame no Murakumo needing a sacrifice was a test over the relationship between the Mikos and that the reason the original Himeko and Chikane suffered after the first war with Orochi pre-21st century was presumably neither were open about their feelings for each other, which their reincarnations in the 21st century finally did so, which convinced AnM to reward them
by allowing them to reunite, depending on which version of the story you're going with.

Also, I learned of one difference between anime and manga: the latter has this narration that the former didn't adapt (quoting the fan translation and because the pages with this narration have Souma, Tsubasa, Girochi, and Kazuki):

"The restored world knows nothing. People don't know the story about the gods and girls. Anything left is myth and folklore - and some nothingness. But as long as there is darkness in people's mind, someone will realize by looking up."

Does that mean (in the manga at least) Orochi has a chance to revive again and seek out its new eight necks, dooming the reincarnated Himeko and Chikane into fighting it and one having to sacrifice the other again?