In this setup, the mafia cannot kill at all. That means essentially, there is no night phase. However, if the Godfather is lynched at any point during the game, the town wins. (This is the only special ability of having the role of "Godfather" for this game).
Another unique quality of this setup is that on Day 1 and only Day 1, if a townie is lynched, that townie gets a one shot chance (like becoming a vigilante) to make a kill. If the Godfather is killed in this way, the town does not automatically win.
Mafia win when there is at least an equal amount of mafia and townies and the town does not have a vigilante kill remaining.