It depends on what the director intended. If the director was framing the shot for widescreen, then that is the best way to view the movie, imho. Older films were often framed at a ratio similar to the tv screen, so you are not missing anything by seeing them widescreen. It's rare that a modern film will be shot in full screen, but Stanley Kubrick did that for a number of his pictures. That's why when you buy the DVD versions of the Shining or Full Metal Jacket they will only come in wide screen versions.

It's unusual that a film that was filmed to be shown full screen would come out in a widescreen version, so I am inclined to think that Godfather wasn't filmed full screen. I could be wrong about this. I think Kubrick's Eyes Wide Shut was shown widescreen in theaters even though Kubrick used full screen when filming it. In an interview he said he tried to frame shots so that they would look good in both versions, but that he preffered full screen. Unusual, but that was Kubrick.