NV basically incorporated the Fellout mod from Fallout 3. It's been more than 200 years since the bombs dropped - the world would absolutely not be that green, nor would it look that destitute. Plant life would be recovering, there would be more wildlife, and the sky would be blue. Fallout 3's interpretation of how the world would look 200 years after a nuclear war is about how accurate CSI: Miami's depiction of real-life Miami, FL is (the sky is not fucking ORANGE in Miami!)
New Vegas is definitely the better game. It's a LOT more stable than Fallout 3 is (probably because Obsidian Entertainment aren't a bunch of lazy sacks of shit) and the gameplay is more interesting. It also includes Hardcore mode by default, which includes a very primitive version of old Fallout 3 standbys like Imp's More Complex Needs, makes ammo have weight (you can no longer carry an armory on your back), and pretty much entirely removes Essential (knocked unconscious by lethal damage rather than simply killed) status from all companions. New Vegas also provides you with a full command system for companions right out of the box, and the companions themselves also tend to be more interesting, each having their own personal sidequest associated with them... even the dog and robot.
The skill system in New Vegas is also greatly improved. Survival returns as a skill, and the Big Guns skill is instead separated and merged with existing weapons skills - Miniguns and their like are governed by the Guns skill, the Explosives skill maintains dominance over Missile Launchers and Grenade Launchers, and Energy Weapons gets the hardware like Gatling Lasers. New Vegas also adds and significantly expands upon the crafting system, allowing you to craft many more item types at a workbench, the new campfire, or the new reloading bench. Guns users with appropriate Repair skill can now recover spent shell casings (as well as buy them from vendors on occasion) and breakdown existing ammo to create new types of ammo at a reloading bench - it's largely relegated to recovering spent ammunition or, with the appropriate perk and Repair skill, breaking down basic ammunition to create much more powerful variants. You can turn standard 5.56mm rounds into 5.56mm Match Rounds (bullets which use the exact same amount of primer and powder in each round and are used for competitive shooting, such as at the Olympics), which increases damage and reduces weapon spread, for example.
You can recycle energy weapons ammunition at a workbench with the appropriate Science skill, and you can also create Over Charge (rounds which cause more damage but also damage the weapon more) and Max Charge (even more damage at the cost of even more damage to the weapon) rounds. You can also convert one type of energy round into any other type - if you have a lot of small energy cells for your laser pistol but find yourself needing more microfusion cells for your plasma rifle, it's easily remedied. Guns users also have this in a limited fashion - each type of bullet requires a specific kind of primer (small pistol, large pistol, shotshell, small rifle, or large rifle), a specific kind of powder (pretty much just pistol or rifle), and the correct type of casing (you'll need .45 Auto shell casings if you're wanting to make .45 Auto bullets, of course.) And this, of course, means that each and every weapon has multiple ammo types available. You can use hollow-point ammunition to shred targets with light armor or armor piercing rounds to make that jackass's power armor worthless. If you're on a budget, you can buy surplus ammo boxes which are easy on the budget but hard on your weapon, or you can buy some .38 Special for your .357 Magnum Revolver to make life easier for your weapon at the cost of causing less damage.
Using a campfire allows you to put your Survival skill to use in the creation of a wide variety of consumables and items. You can make anti-venom, healing powders, create purified drinking water by boiling dirty water, tan gecko hides for higher resale value, or turn that Mole Rat meat into something that's more nutrious and safer to eat. These are all features which were included in many of the most popular mods for Fallout 3, and they made a reappearance in NV.
Also another mod that was wildly popular for Fallout 3, weapon modding is now an included part of New Vegas. Buy a silencer for your pistol or a scope for your rifle. Replace the bolt in your rifle to allow faster firing, or saw off that weapon's stock to reduce its weight. Replace the slider with a high-density version to improve the weapon's durability, or extend the barrel of that grenade launcher to improve its range. Add an advanced targeting system to the missile launcher to make it more accurate, or add a choke to the end of your shotgun to reduce its spread. Lots of possibilities, now possible without having to mod your game.
Another fantastic change is that skill challenges within dialogue are no longer random chance. Each challenge will have a set value that must be met for the attempt to succeed, so there's no guessing involved... and simply spamming quickload until you finally pass that 1% speech challenge is also invalidated. The types of challenges are also widely improved - you can impress storeowners with your knowledge of Guns to convince them to sell their higher end merchandise to you (after all, who wants to waste time selling a $3,500 rifle to some fool who doesn't even know how to sight it in properly?), intimidate enemies with your knowledge of Explosives, or use your Medicine expertise to notice that that guy is being a prick because he was abandoned as a kid. Speech challenges are more widespread as well.
In regards to plot, that's inherently subjective but I've actually been completely drawn in by New Vegas' plot while I honestly didn't give a shit about Fallout 3's. In FO3, you have a clear set of good guys (the Brotherhood of Steel, oddly enough) and a clear set of bad guys (the Enclave, because fuck them.) You're given a couple options to do bad things, but they don't really change anything - you're still forced to be a hero, and if you're the type of person that murders anyone you meet and steals change from children, you're just an anti-hero, I guess.
New Vegas does not have clear bad guys or clear good guys. You are given four different factions to choose from when it comes to deciding how the game ends (one of them being to tell the other three to fuck off and you doing your own thing, which is an immensely satisfying option not present in most games), and none of them are clearly better or worse than any of the others. Mr. House would have you do his bidding, helping him create an autocracy, complete with all the pitfalls of that type of goverment... but the man also has vision and while he might not be "good" in the classical sense, it's hard to disagree with his vision of the future. The NCR is very much like an American-style government, and just like those kinds of government, the power (and money, and resources) is held by the elite few at the top and everyone else is pretty much just a footstool. Caesar's Legion is a glorified army of slavers, but everyone within their society has a place and a purpose, and their territories are widely spoken of as being extremely safe. And while going independent and doing your own thing means everyone is truly free... that also means that nothing will be there to prevent the next band of raiders from wrecking the place, and there will be no set laws a real society could begin to build upon.
It's a lot more of a compelling narrative than "go here and kill these people because they're the designated bad guys."
In regards to DLC, Fallout 3's DLCs are generally pretty lousy because they're largely designed as if the game were a shooter... which, without modding, it simply isn't. Thematically they're all very enjoyable and interesting (with proper modding, Operation Anchorage is a complete blast), but gameplay-wise they're not that great on an unmodded game. Some of them (Operation Anchorage in particular) are also pretty game-breaking in that they allow you to immediately access some of the best possible equipment in the game literally right out of the vault. There are no impediments keeping you from immediately running off to go do that DLC, and just like getting the Adv. Power Armor right away in Fallout 2, it completely trivializes the game.
New Vegas' DLC all share an overall plot, with each DLC foreshadowing the final DLC, Lonesome Road. The ultimate goal of the DLCs is to have your character encounter Ulysses, the Courier who was originally tapped to deliver the item that ended up getting your character shot in the head, and along the way you discover the reason he effectively set you up, and what your character did in the past that caused him to want to do this. You'll also meet characters in each DLC who share relationships with companions and other characters you have within the main game, which offers a little bit of character depth and development the Fallout 3 companions largely lack. Each DLC is visually distinct and has a unique feel to it. Dead Money takes you to the Sierra Madre Casino and the villa which surrounds it - the surrounding area has been covered with a poisonous cloud and is now home to the mysterious, deadly Ghost People. Honest Hearts has you escort a caravan into the pristine, untouched Salt Lake area (I personally didn't like the setting, but I enjoyed the characters and plot.) You'll visit a pre-war high-science think tank in Old World Blues, laden with Atomic Age sci-fi tropes. And finally, Lonesome Road takes you into a desolate, destroyed pre-war missile base as you work your way towards Ulysses and your final confrontation with him. The overall plot and direction of the DLCs is distinct and separate from the main game's plot and largely serves to flesh out the player character's history and the histories and characters of those you interact with. Played in chronological order (Dead Money, then Honest Hearts, followed by Old World Blues, and finally Lonesome Road), each DLC foreshadows the confrontation at the climax of Lonesome Road and fleshes out the character of Ulysses a little more.
Well, and there are two other DLCs which just serve to add items to the game. The Caravan DLC just supplies you with a variety of items at the beginning of the game and I didn't actually bother with it (even though it was just like 50 cents.) The Gun Runners' Arsenal DLC I highly recommend, as it adds a wide variety of weapons and ammo to the game, as well as a variety of interesting and fun challenges, ranging from the fairly easy and/or humorous (kill Caesar by stabbing him with a knife) to the brutally difficult (kill 10 adult Deathclaws with a knife, .22 pistol, or a selection of other very weak weapons.)
As far as modding goes, the list of mods and their maturity is extensive for Fallout 3, while NV's is fairly limited and the mods that do exist feel somewhat immature. On the other hand, NV is still fairly enjoyable right out of the box because Obsidian Entertainment literally looked at the most popular mods for Fallout 3 and put them into New Vegas (minus the fringe things like those stupid nude mods) in one form or another. Both games have UI mods which make the game actually playable on a PC (you know, with high-resolution fonts, proper spacing, and all that stuff), which are the only variety of mod I would say is absolutely, completely required to play - the vanilla games use the same fonts and spacing used for consoles, which is bad news for PCs, which have much higher-resolution displays.
tl;dr: Fallout 3 is garbage, don't waste your time on it. New Vegas is how Fallout 3 should have been.
EDIT: For what it's worth, you will meet Dr. Venture and can have Felicia Day as a companion in New Vegas, while Fallout 3 really only has a few cameo roles by Liam fucking Neeson. Oh, and New Vegas has Muggy.