The competition begins at about 1:35, after they’ve had some words. The song is “Aaye Haaye Dilruba.” Geeta Dutt sings for Helen, Asha Bhosle for Vyjayanthimala; music by S.D. Burman.
P.S. Helen is great to watch in this, but she pretty clearly gets clobbered at the end.
Helen vs. Vyjayanthimala Dance Competition in Dr. Vidya (1962)