From SA Mafia Wiki
As the leader of the Mafia, the Godfather is usually in charge of coordinating the actions of his fellow Mafia members, including final decision of the nightkill target and (if applicable) which Goon is responsible for making the kill. Usually the Godfather will have a chain of command that goes into effect if he dies; the new Mafia leader does not inherit the Godfather's investigation immunity, simply the last word over the nightkill.
The possible presence of a Godfather is often used to cast doubt on Cop investigations that return a town result on a scummy player. On the other hand, sometimes Godfathers play somewhat scummy on purpose in order to draw a cop investigation which they then use to clear themselves by night-killing the cop and confirming his Sanity.