Nope, not the same tune Joe.
Both melodies start high and descend, but otherwise quite different.

Lovely performance here from Paddy Tunney's son John John Tunney sings The Mountain Streams where the Moorcocks Crow who says:

This song has come down to the Tunneys from John's Great-great-grandmother Biddy Travers, who lived on the shores of Lough Erne in the decades before the great Famine.
It is a classic hero roves out with his dog and gun tale, meets and falls for a beautiful girl but she is not really available.
It is one of the really big songs in the Tunney family repertoire and is most closely associated with John's father Paddy who moulded into the wonderful song it is today.
John Tunney, May 2012