Stephen Forsyth, who wrote the song alongside Jim Ryan explains why their song wasn't used:
"During the filming of the James Bond movie Never Say Never Again, I co-wrote the title song for the movie with Jim Ryan. Warner Brothers informed our attorney that the song was to be used as the title song in the picture. However, shortly before its release, Warner Bros informed us that the song could not be used because Michel Legrand, who wrote the score, threatened to sue them, claiming that contractually he had the right to the title song. So my song was never released. The legendary Phyllis Hyman was my first choice to sing the song and working with her is one of the highlights of my musical career. I personally auditioned and sang the song to her while she was having breakfast in her manager’s office. After agreeing to sing the song, she arrived at the studio and, without any rehearsal and only having heard the song sung once at the breakfast audition, sang the song in one perfect take. Phyllis sadly took her own life in the early nineties. The year before she died, she called me late one night and told me she felt that "Never Say Never Again" was her best and favorite recording."
While I believe this is the superior theme, it would have worked much better over a traditional opening title sequence.