Babe is a term of endearment. It is sort of like honey or sweetie. You say it as sort of a nickname to someone you adore. For example someone may call their wife babe. Baby is a noun used to a small child who was just born. Although, you can also use it in the way of endearment. Baby: I just gave birth to a baby. Or I call my significant other baby. Babe: I call my significant other babe.
That is because baby and babe can both be used in the same way. It doesn't really matter if you use babe or baby in songs talking about others. It is sort of just a cutsie nickname. I guess they could decide which one to use in songs depending on the beats and rhyme of the song.
They both mean a human baby and a term of endearment. Although, the use of the word "babe" to describe a small baby is outdated. "Baby" and "Babe" are used amongst couples much in the same way you would say "honey", "sweetheart", "darling", "dear".