Barry Mann and Cynthia Weil originally wrote "On Broadway" for a group called The Cookies. Their friends, Carole King and Gerry Goffin, were writing for them and they were short one song. The Cookies cut it first, but it was not released until the 1963 compilation, The Dimension Dolls - Volume 1. The Crystals cut it second, but it was issued earlier, on their 1962 LP, Twist Uptown. Peter Stoller confirms this in an earlier post.
After these two recordings, Mann and Weil's publisher told them that Jerry Lieber and Mike Stoller were looking for songs for The Drifters. All four worked together on the re-write, which was more than just "rhythm and harmonies"; the lyric and melody also changed.