Seeing as I've owned and beaten all the game in the Fallout series (not counting Tactics, we never speak of Tactics), I can formulate a pretty decent comparison between them all. Especially when it comes to how "Fallout-y" they feel.
Fallout 1+2 are my favourite games in the series. The stories were brilliant, the characters were memorable and the sense of being in a post-apocalyptic world was there 100% percent of the time. Even though I know a lot of people had no real love for the combat system, I loved it. Largely for one reason that neither of the two most recent ones implemented:
True body part targeting. In 1 and 2, you could aim your guns not only at the head, legs, core and weapon (like in FO3 and NV) but also the eyes and groin. Those shots were hard to pull off for anyone but a skilled sniper, but were they ever worth it. Shots in the eyes either outright killed some or at least blinded them. Groin shots were great for highly-armoured opponents. Why? Because no matter how much armour you have, a .223 round fired from a rifle is going to hurt a lot when it impacts your ghoulies.
Oh, and you could aim melee attacks the same way. Which was great.
Fallout 3 was a huge departure from the first two games, keeping only a few things intact (namely the title, a few weapons, the Vaults and little things here and there.) It could just of as easily been an entirely different game franchise, and no one would of been the wiser. The Fallout name helped carry it along, despite it have as much in common with the original games as a shrew and human have in common.
I will give it one thing, though. They managed to get the sense of a post-apocalyptic world pretty darn well. There were large expanses between the towns, and those towns were ramshackle at best.
Not to even mention the story.
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SPOILERS FOR FO3 PLOT
The story started out so well! You grow up in the Vault, grow attached to the character of your father then one day he bugs out. Why? Who knows! Let's go follow him and find out! After far too long of an story arc, you find out neither one of you were born in the Vault. He just managed to somehow get in after your mother did during birth. And his plan? His dream? Create fresh drinking water for the Wasteland. Noble, yes, but not something to get really excited about.
Speaking of good ol' Dad, hope you never really got attached to him. Even though you were supposed to. You only end up seeing him a few times after the tutorial part of the game. Then he dies for no reason.
Great, thanks game. The whole reason for me leaving the Vault, killing countless people and creatures while marching up and down the Wasteland is moot. You find him, and he just up and dies.
Then when you get to the very end of main story, finish the final quest...Credits roll. That's it. No more roaming for you! You beat it and can only restore to the most recent save. Worse decision by developers evar.
END SPOILERS
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Now, on to New Vegas.
New Vegas is a really mixed bag for me. The feel of being in a destroyed world is only sorta there (I blame the large, developed towns a New Vegas for that). STALKER has a better sense of that, for crying out loud. It kinda ruins some of the magic.
However! They did improve on a lot of things. Even though at least one of your shuns the idea that it is more Fallout-y, I agree with that statement. The story, music and other things remind me far more of the original two than FO3 ever managed (I'll go into more detail about this if requested). That is not saying much all things considered, but it is still nice.
I must say that playing through vanilla New Vegas once before the DLCs came out was a little bit of a drag, but I still enjoyed it. Yet when all the DLCs were finally out (and I was playing a super-modded version of the game), I had more fun that I thought was possible. The DLCs give some nice throwbacks to the lore and previous stories, and provided some interesting albeit highly ambiguous back story for your character. That combined with all the little tweaks and additions from the mods makes for one of my favourite gaming experiences.