Alright, so you wanted some examples of how New Vegas is different than FO3, and why this makes it a better - and even a
good - game. This is going to be a bit long-winded, so feel free to tl;dr and move on. There will be also be minor-to-not-quite-major spoilers, mostly on a few specific quests and some general setting things. Nothing about the end of the main quest, because I've personally not yet finished it.
To start with, I judge and define sandbox RPGs like New Vegas by their setting, atmosphere, and quest design. Obviously the first two are largely tied together, so I'll address them as a single subject. I
won't be addressing the "improved writing", because I'm admittedly not particularly good at judging one piece of prose vs. another. I will say there haven't been any lines that made me facepalm ("Steel be with you!", "I'm looking for a middle-aged man"), but there have been a few that came across as particularly amusing or witty. However, I largely consider the vastly improved setting and atmopshere to be a by-product of a generally more talented writing team.
So what's good about New Vegas' setting and atmosphere? Obviously they're tied closely together - the setting feeds the atmosphere, and vice versa.
Let's start with the NCR - perhaps the "main" faction in the game, around which a lot of the politics of the others circulate. Any notion that the NCR are the clear-cut "good guys" of the story is put to rest fairly early on, once you learn the Powder Gangers - the first criminal group you're likely to come across - only exist because the NCR fucked up and allowed a major prison break to occur, even
losing that prison to the group in the process. The more you explore the world, the more it becomes clear that the NCR is anything
but the stabilizing force it wants to be.
Caravans from the west are being blocked at the NCR border because the NCR simply can't control the roads - evidenced by the random raider groups along the road, and the Deathclaws northeast of Goodsprings (the starting city) that help discourage traffic in that direction. This is only further emphasized when you reach Nipton - a city quite deep into NCR territory - that's been sacked by the Legion. It's clear from almost the onset that those in the NCR might have good
intentions, but they simply don't have the manpower to back that up. Further investigation reveals that this is largely because the "Hoover Dam campaign" is growing less and less popular back in California, so it's harder and harder for new troops and supplies to show up on the front lines. Morale is shit, and for good reason.
It's all rather tragic, because it's implied the Legion has numbers
far greater than they've shown, and that if the NCR loses at Hoover Dam, all of California would probably be next. Finally, you eventually learn that the major victory the NCR had over the Legion a few years back happened not because the NCR had the numbers, morale, and firepower, but rather because they tricked the Legion commander at the time into doing something stupid - something they certainly won't fall for when the Legion comes back.
There's things you learn about the NCR here and there - the "Bitter Springs" incident, for instance, which is gradually revealed and ties into backstory of one of the companions as well as one of the other major factions. In fact, if and when you
find Bitter Springs, it's become nothing more than a shoddy NCR refugee camp - going so far as to hang its flag upside down to indicate "distress" - and you can even see the old graffiti from the previous faction half-assedly covered by NCR signs and propaganda. One of the quests there is to deal with a member of the wronged faction who is still taking his revenge on the NCR, even years later.
The Legion, of course, are the "silly Roman cosplaying slavers" - whether you like them may depend largely on if you buy the reason they exist, which I personally did. They largely serve as the game's major threat, but like all factions, they have their own motivations and backstory. One of the companions has a major (bad) history with them, and just having him in your party makes them all hostile. He warns you when you get near certain Legion strongholds that he plans to go in shooting, leading to a great exchange:
Companion: "If we go in there, I'm going to kill every Legion member I see. Is that a problem?"
Player: "That's not a problem, that's a solution."
Then there's the smaller side stuff - the skyway-scoffed Black Mountain/Tabitha thread ties into the greater Super Mutant side story. Black Mountain was a Super Mutant community run by Marcus, who was run off by Tabitha. He founds another community - Jacobstown - in an old ski lodge - who have their own problems dealing with the gradually-more-insane Nightkin. Marcus has promised them a cure, but so far hasn't been able to deliver, and they are getting restless. If they leave, it means groups of pissed-off Nightkin will be roaming the Mojave.
The Brotherhood of Steel is obviously aroud - the FO1/FO2 Brotherhood, thankfully - but they've been driven into isolation after a devestating defeat by the NCR over Helios One. It might be surprising that the Brotherhood could lose to
anyone, but the NCR kept pouring man after man into the battle, and the BoS gradually ran out of supplies and had to make a run for their bunker. Their leader has been keeping them in lockdown, fearing that the NCR has only grown stronger since the Brotherhood was defeated - but throughout the course of their main questline, he learns what the player already knew - the NCR's "hold" on the Mojave is tenuous, at best. Of course, there's a side thread where the player can work with a faction who is opposed to the Brotherhood's current "lockdown", and you can even have the leader replaced - which leader is in charge after the quest line directly impacts what the
next quest is (the other option isn't available for the rest of the game).
One of
those quests involves wiping out a minor faction - a faction that also has a quest for you, which you may never get, if you kill them.
That faction's quest can't even be completed without sacraficing one of your possible companions. At that point, you either have to give her up, or killing them all.
The Followers of the Apocalypse have been following the NCR's expansion, more or less trying to clean up the mess the NCR leaves behind it. They don't see eye to eye with the NCR or the Brotherhood - the second becoming all the more clear during one of the possible endings for one of the companion's quests. They aren't particularly interesting as a faction, but their interactions with NCR leads to choices about the good of the NCR vs. the good of everyone
but the NCR.
Finally, there's the wild card - the titular New Vegas itself. Mr. House has his own mysterious agenda (which I honestly haven't uncovered yet), and, of course, the Families under his "command" aren't entirely thrilled being the underdogs. Vegas is a problem for the NCR, because it's not under their control, yet it's a constant drain on their resources. To control it would be a major win for the politicians back home, but right now, it's little more than a necessary evil - the NCR simply can't afford to take on Mr. House with the Legion threatening them on the other side of the Colorado. Finally, the recent "revival" of Vegas by Mr. House has forced the tribal chaos of the old Vegas into the surrounding areas - most notably in Freeside, but also in Westside, North Vegas, and Red Canyon.
There's more I haven't touched on, but this post is long enough as it is, so there you go: the setting and atmosphere of New Vegas, which (I hope) easily shows how these areas are vastly superior to Fallout 3's disconnected "theme park" areas and factions. I've also touched briefly on the interconnected quests with multiple solutions, but if people want another rambling post, I can go into more detail regarding a specific questline.
