Are Buddha nature and Original enlightenment different, or always identical?
Yes.  The difference in terms just highlights a difference in emphasis.  All zen teachings basically point to the same thing.  The difference is dependent on what obstacle the teacher is pointing too.  For someone who is always striving, a teacher might call it original mind to emphasize that he is already in possession of what he's striving for.  For someone who thinks zen is about rarefied states or accomplishments, a teacher might call it ordinary mind.  For someone who is caught up in intellectualizing, he might simply hold up a flower.
Its all the same shit - namely the shit you are caught up in.