The beauty and horror of HBO is that everything airs repeatedly, especially with the satellite HBO package, which has seven HBO channels. When they air Serenity or Thank You For Smoking, it's a blessing, but pure torture with Date Movie. Over the last few days, I've had a chance to watch the Sopranos finale two or three more times, and I think I understand the ending much more clearly than before.
The key is the very beginning of the diner scene. When Tony first walks into the diner, he sees himself at the booth, and he's dressed differently. He comes in wearing a drab gray shirt under his leather jacket, looking frazzled, but at the booth he's wearing a different shirt [update: same shirt] and looking rather normal and relaxed. That's the setup that tells us what happens in the rest of the scene is a fantasy, lived only in Tony's mind.
What does Tony fantasize about? A normal family life. His son has become well-adjusted. His wife seems happy. But reality begins to intrude; Carmela starts talking about Carlo, and Tony notices the man wearing the Members Only jacket nervously.
The entire series has been about Tony's attempt to fantasize himself as a normal family man. Since the first episode, Tony sought therapy as a means in which to resolve the conflict between his fantasy and his reality. He lived in denial of his true murderous, sociopathic nature for the entire arc of the series, abetted by Dr. Melfi, who actively tried to help him do so. He wanted his belief in his supposed goodness and normality to become his reality.
Instead, in this fantasy, Tony finally realizes that he's deluding himself. David Chase throws in a red herring with Meadow and the parallel parking, but otherwise it's all about Tony's delusion crumbling. His daughter succumbed to his fantasy, giving up a career in medicine to defend people like Tony, whom the FBI supposedly persecute because of their Italian descent. His son now works for him and Little Carmine. His wife openly discusses the one brutal piece of reality that he most wants to forget -- Carlo, the guy who's probably going to put Tony in the joint for the rest of his life.
It's not for nothing that fantasy-Tony plays "Don't Stop Believing" at the jukebox. Tony has held tight to this fantasy of normality for years. It's been the root of his depression and probably of his panic attacks, which started when he began the life; recall that Tony's first faint happened when he was supposed to go do a heist with his cousin Tony Blundetto. His fantasy self is telling him not to end the delusion.
But that's exactly what he does. Tony doesn't get whacked in the final episode; Tony just kills the fantasy. That's what the abrupt ending means, with the song cutting out at "Don't stop --". The delusion is dead -- and Tony finally has to face reality about who he is and what is in store for him. The series' main story arc has come to an end.
And that's an ending that befits the series.
UPDATE: The First Mate had an interesting interpretation of Meadow's parallel parking in the final fantasy sequence. She has been pulling away from the family at times, and at other times, in the same kind of denial as Tony. The back-and-forth of the parallel parking in Tony's mind might reflect her vacillation, but in the end she joins the family fantasy. That might be what brings Tony to end the delusion -- once she rushes into the diner, it abruptly ends.
UPDATE III: It could be the same shirt, as commenter mscala believes I've watched the sequence in slow-mo a few times, and it looks like a different shirt to me, but I could be wrong about that detail. I don't think it undermines this analysis, though.
UPDATE IV: I've added the YouTube for the final 4:49 of the show. Tony walks into the diner, then the camera cuts to Tony's perspective staring at an empty booth. It cuts back to Tony's face, and then back from Tony's perspective again, with Tony sitting in the previously empty booth. It's a signal that we're in Tony's fantasy.
Here's another series of clues. When Tony is waiting for Carmela and the family to join him, the camera focuses on three song selections on the jukebox. The first is "Who Will You Run To/Magic Man" by Heart. The second is "Don't Stop Believing/Any Way You Want It" by Journey. The third is "I Gotta Be Me/A Lonely Place" by Tony Bennett. Tony chooses the second choice in his fantasy rather than the third; he doesn't want to be himself, which would certainly put him in a lonely place. Unfortunately, Tony stops believing.