I'm going to try to limit my length to 1,000 words (not counting the 18 in this sentence.)
If you played Fallout 3, you've played New Vegas but you might be disappointed. New features provide a more challenging and balanced experience but the game doesn't feel as lovingly crafted as its predecessor. If Fallout 3 was a 9/10, I'd say New Vegas is an 8/10.
New features and balance changes are numerous. There are more ammo types, more guns, new skills, new perks, new enemies, and faction reputations to watch. A hardcore mode makes follower death permanent, gives ammo weight, and forces you to eat, drink and sleep just like a real boy. Bobblehead stat bonuses are replaced by implants that are both expensive and limited by your endurance. Skill books are rare but offset by more common skill magazines that offer a very short boost. Guns have both a skill and strength requirement. Broken items like the dart gun and stealth suit have no equivalents. I've played as an Int/Cha focused character and a Str/End focused character and I'm very pleased with how the shift in focus completely changes the game. In Fallout 3 it didn't seem to have such a pronounced effect.
Hardcore mode is an interesting addition but more of an annoyance than anything else. It makes all healing take some time which makes combat more tense, but you carry so much food and water to offset hunger and thirst that most of the time you'll just eat a ton of junk since stimpacks are reduced to "food that doesn't fill you up." The hunger/thirst situation could be a serious concern for low Str characters, but my 10 Str brute has enough food to feed the entire Capital Wasteland so this wasn't balanced very well. Ammo weight is so insignificant and the need to carry varied weapons is so little that you'll rarely bother limiting your ammo due to encumbrance unless you plan on only using missile launchers. Limb damage can only be restored by a relatively rare item or doctors; I find this to be ridiculously inconvenient as it means I sometimes have to leave an area half-explored because some enemy got lucky with a headshot.
Unless you plan on gambling and winning caps are much tighter in the game. This is a problem because I haven't noticed many of the high-end weapons showing up on enemies so you have to purchase them with caps. It's nice that this clashes with the desire to boost your stats with implants but it's also frustrating when you hit level 30 and still can't afford the gun you want. I feel they went a little crazy with ammo variety and included too many. There's a few ammo types that seem too rare; across 2 playthroughs I was able to acquire 50 rounds. No way am I going to spend 8 quests' worth of caps on a gun that's only good for 50 rounds. I haven't used energy weapons but the opposite seems true for them; there's maybe 4 standard ammo types per 1 energy ammo type and by the end of the game I've got more microfusion cells than several different standard ammo types combined.
The challenge is much better in New Vegas. As the non-brute character I found myself in need of a follower in many combat situations. Some of the enemies are probably too difficult, but in many ways I feel there *should* be some enemies you prefer to avoid. VATS accuracy has been dialed down enough I don't use it much; gone are the days when it was more reliable to use VATS than manually aim. The only thing that hurts here is the follower AI is atrocious. You might see a dangerous enemy far away and tell your follower to be passive, but that won't stop them from breaking out a melee weapon and attacking anyway. This hurts more in hardcore, where follower death is permanent. I sometimes have to move my followers somewhere else and tell them to wait when facing something that will require finesse.
Overall I'm pleased with how many different ways you can finish quests. If you have high charisma and speech you can smooth talk your way out of many fights. If you have high intelligence or science you can figure your way out of many fights. If you have neither and are focused on combat you can shoot your way through many fights. Occasionally the "no smarts" approach requires a cruddy "find these items" solution, and I wish they had found a different approach.
This brings me to the ugly part: what's missing from Fallout 3. I feel like the soul of the game just isn't there. The Capital Wasteland was at least 50% bigger and mostly desolate. In NV the map is smaller and there are twice as many locations, but there's half the story associated with them. Many locations are simply a map marker, and many more are just small buildings with insignificant loot and uninteresting enemies. Nearly every location in Fallout 3 had a story you could unravel via terminals, holotapes, and items left behind. NV's locations are as desolate and soulless as FO3's wasteland. The randomly spawned raider encounters from FO3 aren't present; this puts a cramp on what you can do for caps. Most of the quests have a very grindy JRPG "do something the NPC is too lazy to do" feel to them, and often you're only doing them for XP or caps. I feel like FO3 told a much better story and created a better universe than NV.
And then there's the glitches. My first playthrough was nearly ruined by a follower glitch that made none of my followers use their appropriate weapons. Your weapon will often glitch into the center of the screen and obstruct your view when it's not busy falling through the bottom of the screen and returning from the top. NPCs will become hostile for no reason. Quest NPCs will decide to not interact with you. If you were used to the glitches in Fallout 3, they're worse in NV. This is unacceptable and only in the video games industry can such a situation lead to a profitable game.
If you were a fan of Fallout 3 then NV is worth at least one play. I think Fallout 3 is a much more enjoyable game, but it has balance problems that make certain character builds dominate. If you're looking to see what Fallout is all about I think you should start with FO3 and only move to NV if you really enjoy it.
(~1,080 words; not bad considering I usually only stop when I hit the 10k character boundary!)