They also bent-over-backwards to assuage the French in North Africa. But that's a book-length discussion right there.
The big reason why the Mafia cooperated with the Allies in Sicily is that the Mussolini regime cracked down hard on the Mob in Sicily. I would have to do some research on this but I believe there was one Mussolini official in particular who was especially hated by the Mafia. Also keep in mind that almost every family in Sicily had relatives in America. When the Americans landed many of the troops could speak to their paisanos in their native dialect.