Why is a film directed by an Auteur Director supposed to be better? For me, it's because in order to be considered "Auteur" as a Director, you have to have a grasp of all aspects of film and Cinema, but to a very high degree. You have to fully "Get It". You have to rise above the solidly capable, and venture into that scary land of the Artiste. It takes understanding of the form, genius, fusion, clear focus even when it appears to be unclear.
I grew up watching PBS, Channel 9 in Tacoma, Washington in the 1960s. I don't know who was responsible at that station, but they did a bang up job.
I had no idea what I was seeing but in comparing it to most of the American films I was seeing, there was something special going on there. I grew up going to the Drive-In Theater every Friday night as our Step-Dad was an Asst Manager. It was his night job, the job he enjoyed. Having been a big band leader, it was his chance to get out of the warehouse he worked in during the day, put on a suit and meet the public; even if it was only to sell them tickets. Later I too worked there at night all through High School. So I grew up seeing a lot of films.
On top of all that, my mother saw Hollywood actors as our American Royalty and so we celebrated when we watched the Oscars. Film, is kind of in my blood. And so it makes sense that I would lean toward the top end of filmmaking, the Auteur films.
So, what exactly is "Auteur"? The technical definition of the term "Auteur" (French for author) describes film directors (or, more rarely, producers, or writers) who are considered to have a distinctive, recognizable style, because they:
(a) repeatedly return to the same subject matter,
(b) habitually address a particular psychological or moral theme,
(c) employ a recurring visual and aesthetic style, or
(d) demonstrate any combination of the above.
In theory, an Auteur's films are identifiable regardless of their genre. The term was first applied in its cinematic sense in François Truffaut's 1954 essay "A Certain Tendency of the French Cinema" - Wikipedia
Using a list of Auteur Directors, I picked the ones that were influential starting from before I was a teenager to adulthood and after I had spent some time studying Film Theory and Cinema at my University.
I would argue that some of directors on the list I was looking at, I really wouldn't consider Auteur, but well known, proflific, or highly competent. I have a more stringent definition for Auteur, similar to my differentiation between Artist and Artiste. I have always tried to differentiate between those highly competent and prolific Artists and Genius Artists, or those individuals who took their craft or art to a much higher degree. Thus I see Director and Auteur Director, Artist and Artiste, for wont of a better term for either one.
Here is my personal, mostly complete list, not in any order. I may have left some off, but I think I got most of them, those who have touched my life in some way. I may have included some perhaps through emotionalism, and who could arguably be removed, but not many I think.
Rainer Werner Fassbinder
Georges Melies [500 films to his name, he destroyed them all]
Guillermo del Toro
Paul Verhoeven (mostly for his preHollywood films)
Jean Luc Goddard
Sir David Lean
Sergei M. Eisenstein