Pavane

It's very common for a broadside to use the song of an old song for a newer one (saves the buyer having to learn the tune from the street singer or make one up). See, for example, the earliest broadside versions Black Velvet Band where the tune is given as "Tars of The Blanch". I think that the simple answer to your question about the relationship between the two songs is that they were to the same tune. Period.

If you're interested in looking at the wider issues associated with the origins of this song, there are quite a few earlier threads. However, I don't recall the Green Willow version as being at all popular until Steeleye recorded it. The version I used to know was an older version of this one In Her Hair She Wore a Yellow Ribbon and it might be worth looking for it under that name or a similar one.

Cheers!
Ian