Cash's version and Reznor's interpretations are and were radically different "covers" with different themes, ideals and perspectives they were trying to get across. Clearly, most music fans are more aware and cognizant of "Hurt" being Cash's song then Reznor's. That's why Restore has said many times since Cash's death that it was as much his song and NIN's.
Cash always had a negative, pessimistic view of himself as a bit of a hypocrite, philanderer, rock star who abused drugs and alcohol and was never shy about calling out or reminding viewers or his fans that he wasnt such a angelic, great human being, even though legions of fans and American public saw him as an upstanding, decent-to-good human being who had had public issues and troubles, but had overcome them. Maybe his cover of "Hurt" was also trying to convey to listeners that they should be cautious and mindful about wanting to meet and get to know their heroes, because there's a chance they could end up disappointed or disillusioned.