I actually first played this game on the Wii! This game and its sequel were developed by EA Redwood Shores, who eventually changed their name to Visceral Games -- AKA, the studio that gave us the Dead Space series. For a time, they were responsible for some fantastic entries in the movie tie-in genre, such as it was, giving us The Godfather I and II, as well as the beloved The Lord of the Rings: Return of the King.
The Godfather, like a lot of these games on the Dreamlist, was not necessarily the top of the mountain as far as mafia games go. It was, however, a very strong attempt to create a parallel narrative to the film, with you taking on the role of a recruit in the Corelone family. Several of the original voice actors reprise their roles, and even Marlon Brando himself recorded lines for what would be his final role. (Unfortunately, due to his failing health, Brando's lines were deemed unusable and replaced.) And the story, from what I can remember, is very good, especially if you're a fan of the films.
I would say that The Godfather is to the Mafia series what Saints Row was to the GTA series. It's a very creative alternative, but lacking the overall strength of its competitor. A comfortable 2nd place, but that didn't make it any less of a good game. It's from a unique time when we got some truly great movie tie-in games with a lot of work and heart put into them, most of which vanished with the next console generation.
I'd love to get the chance to play it again. It's a Godfather game, complete with a beautiful open world, a great story, and some of the original actors. It may not have been Game of the Year material, but it didn't deserve to vanish.