The Sumerian and Akkadian versions have some very striking differences, including who goes down to fetch Inanna (or Ištar) from the underworld. I think the same basic idea is the same, though: Asûšunamir is between conventional genders and so, being already liminal, can cross the boundaries between worlds all the more easily. Same for the kur-gara and gala-tura that Enki creates in the Sumerian version, I would imagine.

The later portions with Dumuzi-Tammuz are also much more detailed in the Sumerian version, while the Akkadian text is not only fragmentary in terms of the manuscript but ritualistic and terse in language, presumably because the audience was already familiar with the story and didn't need all the details spelled out. Great fun for the translator in whose culture this myth is not in fact embedded . . .