"Clinton" was actually an old name for "Hell's Kitchen" that came back. Like Puzo's childhood neighborhood, this was Young Vito's stomping ground in the book, but FFC changed it to Little Italy in the film.

http://hellskitchen.net/develop/news/jf040995.html

While there's general agreement about the community's rebound, there's less accord about what to call the area. That's because its original name, Clinton, is now used interchangeably with Hell's Kitchen.
Real estate brokers say there are subtle distinctions between the names. "If it's a rental, you use Hell's Kitchen," said Patrick Lilly, who owns a real estate brokerage bearing his name. "If it's a sale you use Clinton."

(The district is named for George Clinton, the first New York Governor, who owned a vast tract in the area in the early 1800's. Local legend has it that its sobriquet comes from a policeman's remark on a blazing day in 1881 that it was "hot as hell's kitchen.")