Laura,
Nope, not a different song all together, just one of the many variants that have cropped up over 500 years. At the website I set up for collecting verses (clik.to/frog ), I also created a page for tunes and a page for refrains. There are a surprising number of tunes with the same basic words. In the US, 'gammon' usually becomes 'cabbage' because I don't think anyone on this side of the pond calls bacon 'gammon'. Also, from what I've been able to dig up, Anthony Roly (or Rowley) was probably a reference to an Italian opera singer that made a splash in London in the 18th or 18th century. But I've not been able to get anything definitive on this.

If that link doesn't work, try
http://home.earthlink.net/~highying/froggy/froggy.html

all for now
david