I think it represents the curiousity of man. The thing that separates us from all other beings on this planet happens to be the thing that creates our fallen nature. God did it as a test, and instead of having faith, well Adam and Eve chose to disobey their Creator and give into their human traits.
In order to be one with God you'd have to be like God (pure soul, etc) meaning once they shunned him with their curiosity, they could no longer be part of Him, as they were distinctly human ie imperfect beings
Many prophets and messengers were sent before the biblical prophets (Abraham, Moses, Jesus etc) so
there is a good chance the Sumerians got this story from some prophet before the ones mentioned in the bible.
For example, Gilgamesh epic contains the story of Noah and the flood aswel.