Well, what I consider to be a true Fallout game has already been summed up by the rest of the posters here; Fallout is, at its core, a role playing game--ever since Black Isle started the franchise, that's always been one of its key features. What Fallout 4 seems to forget is that, while you can give us a starting goal and even a backstory, you can't make it consume the Character.
When we played the original Fallout games, there wasn't a lot of interpretation to your characters past. One was a Vault Dweller, the other was a Tribal. But you had complete control over their personalities; were you a cruel and ruthless murderer, or were you more of a wielder of words? The original games and NV all had some amazing outcomes to their quests that I honestly didn't expect! Hell, I didn't think the Khans were going to get any mention when I beat the first game, but even a seemingly tiny quest like that got mention in the end credits!
No matter what path you chose, at the end of the game your choices mattered. Make the wrong choice and the NCR would never have become to force it is today, or the Master's Super Mutants would raid Vault 13 in what is probably my second favorite non-standard ending in the Fallout franchise behind siding with Father Elijah.
In Fallout 3, Bethesda took out some of those key elements. It didn't matter if you blew up Megaton or helped the Ghouls kill everyone in Tenpenny Tower, very few people would even know it happened at all, or for that matter care. Another thing Bethesda took out was that Fallout, in my opinion, was never meant to be a game about surviving a Post Nuclear wasteland. It was about rebuilding what was lost in the war so many years ago--picking up the pieces and making something new.
Fallout NV, as most of us will agree to, was far more faithful to the franchise in this regard. The world wasn't a desolate wasteland sitting in stagnant rot for 200 years like DC was, it was actually becoming more civilized--even if, in the Legion's case--that civilization was being brought on through bloodshed. The war has been over long enough that people have the desire and the means to rebuilt. To quote Dead Money's catchphrase, the series struck me as humanity "beginning again."
Fallout 4... um.. it was about shooting things, I think? With not choice or consequence? The game touches on the rebuilding civilization part, but completely cuts every other aspect of the franchise out of the equation. At the end of the day, Fallout 4 was just a FPS wearing Fallout's colors.
So to summarize, Fallout is, to me, about choices and consequences and rebuilding what was lost. And, in hindsight, the wacky humor is a welcome staple Bethesda needs to work harder on.