I think it a Corleone family's strategy to make themselves look weak before their enemies.
Vito,by giving money to Fanuci,makes him believe that he had no "bad intentions" against him,that's why Fanuci walks around alone,careless.
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Michael was even more clever than that. But giving Fanucci only a part of the money he demanded, and telling Fanucci that he was a little short, out of work, needed time, "let me owe you the money," Michael was making Fanucci think that maybe Fanucci could find work for Michael, exploit him, etc. In other words, making Fanucci relax, be confident, not expect what happened.
What the film doesn't clear however,is what was the reaction by the Fanuci's bosses.Clemenza says that the neibourhood was "given" to Fanuci by the "Black Hand" or something.No reaction by the organization however...
As we saw, Michael always doubted that Fanucci was "mobbed up." When Clemenza tells him that Fanucci has "a license from Maranzalla" to work that neighborhood, Michael doubts it. Why? Because, as he tells Tessio and Clemenza, he knows two neighborhood gambling operators who don't pay Fanucci. And, in a deleted scene, he sees two punk kids attempt to cut Fanucci's throat. Fanucci kills one, but lets the other pay him off. Michael tells Genco it's a sign of weakness: "In my village, if you attack someone, you'd better finish him off."
Constantino, since you're from Greece, you should know that, in American cities in the early years of the 20th Century, immigrants were often extorted not by big organizations like the Mafia or the Camorra, but by individuals or small gangs who called themselves "The Black Hand," or some other fearsome title that made their victims think they were part of a much larger organization. The novel points out that, after Fanucci's murder, no one was collecting from the businesses and gambling games that Fanucci was extorting--another sign that Fanucci was a solo operator, and really didn't have "a license from Maranzalla." That's why Clemenza and Tessio convinced Michael to step into that vacuum and become the new boss of the neighborhood.
