I would say that one of the primary differences between the Classical Hollywood style of movie making and the European style was that the latter was out to make a social, political, and/ or artistic statement about being in the world.
Artists like Fellini and Truffaut saw themselves as "artists" and believed that their cinema should highlight this with uniqueness in aesthetics and approaches to making films. This was not the primary motivation in the Classical Hollywood setting where making movies was a business. The idea of "If you want a message, go to Western Union," was something that drove the executives and the movie making industry in the Classical Hollywood setting. In this realm, "continuity editing" was embraced, meaning that there was no attention being called to the technical elements of filmmaking.
This was not the case in the Italian neorealism and the French new wave, movements that actively sought to bring attention to innovations and new technical approaches to the craft of making films. The notion of a cinephile was embedded in the idea of Italian neorealism and the French new wave, a paradigm shift from the business and economic approaches within the Classical Hollywood approach to making movies.
I think that one of the most significant differences between the Classical Hollywood style of filmmaking and the European New Wave was the role of philosophy in the construction of art. The European New Wave cinema in the form of the French and Italian movements really sought to use film as a medium to bring out philosophical ideas about art and consciousness. In the star- studded medium of the Classical Hollywood construction, this is not the case. The notion of being able to make a film that will entertain and make money is the underlying element to all Classical Hollywood films. This difference is significant as it underscores much about films and how they are made. The Europeans understood film to be a dynamic medium where statements about art and philosophy can emerge with the director being an "autere" regarding themselves and art along with what it means to be a human being. Actors are philosophical extensions, extrapolations and embodiment of a set of ideas. For example, existentialism and Sartrean ideas could be put to film. This would not be the case for Classical Hollywood, which would never green light a production which for two hours talks about human agony in the lack of transcendence. In the Hollywood sense, this is not the purpose of film, as much as it is to generate entertainment values for a couple of hours and turn a tidy profit for huge studio magnates who are able to pay actors a pittance for their work while the studio represents the alpha and omega of cinematic creation.