I don’t know anything about music, so sorry that this will sound a bit ignorant.
What do you call vocals being used as part of the tune, rather than being sung along to the tune?
I admit - you have me completely baffled. 
Do you mean the singing without the use of actual words? In classical music, vocalisation and the related noun vocalise is the term I’d use. But I suspect that in different genres, other descriptions may be better.
Related verb, I mean (better link)
An example would be worth a thousand words.
Scatting ? Like in jazz?
As an example, I’ll use Paul McCartney and Wings.
“Mull of Kintyre” does not have the sort of singing that I’m talking about. The singing in the song, while it adds to the song, is not an integral part of the listening experience.
If you contrast that to “Listen to what the Man Said”, bits like “Well I don’t know but I say love is blind” or “Leaves behind a tragic world” markedly change the character of the song from how it would be without the vocals.
Indeed. If you are thinking of something like “Great Gig in the Sky” from Dark Side of the Moon, then I’m pretty sure that’s vocalise (and it does work as a noun, also).
I think the OP is talking about when the melody line is carried significantly by vocals (not necessarily worldess mouth-music - actual lyrics), rather than the vocals following a conspicuous instrumental melody.
Sounds like the difference between melodic singing (“What The Man Said”) and, what? What would be the opposite. Prosaic singing (e.g. Beatles “All You Need is Love”, Cake’s version of “I Will Survive”)? I think the word “discordant” to describe unmelodious singing is a little too strong.
That would pretty much cover the vast majority of pop music. Most of the time, there is no instrument doubling the vocal melody. The band usually provides rhythm, harmony, and sometimes countermelody, but the vocalist provides the melody.
I don’t think there would be any specific term for this.