The NY version with Kelsy Grammer follows the movie very closely. I'm not sure if was a play first but it certainly could have been. It has very few stages and settings and characters so it probably was a play first. I'm surprised Nathan Lane didn't do the play first!
The movie was created prior to the play and as a result the play follows the movie very well. As with all features presented across different mediums there will be some differences between the two.
