To be honest, I like both games. But each for very different reasons.
In my opinion, Skyrim has the better setting. Post-Apocalyptic has been getting a little dull for me as of late. A northern land filled with dragons, magic, war, and vampires? What's not to like? I'd like to live in that setting given the chance, there's also the fact that there are several towns filled with people that offer various quests. Rather than a "town" with a casino filled with about 3 unique NPC's and a hotel. I like that there are actual towns filled with unique characters in skyrim.
However, while Skyrim has the better setting(my opinion) NV does a lot more to increase your immersion. With the static-filled radio broadcasts to the gritty caravan/gang-battles I always seem to come across. Your decisions actually affect things in this game. However, I very, very, very much hate the fact that once you beat the game the credits roll and you can only reload one of your previous saves and live in the New, New Vegas that you created.
The combat was very much the same In my opinion, considering the games use very similar engines, with a few variations. Like VATS, in new vegas. Magic/Shouts/Perks in skyrim. I feel that skyrim would have the advantage if it weren't for the awesome fact that in NV you can blow peoples legs off, disable them. Or whatever the fuck. New Vegas gets the point here, In my opinion. You just have a much larger pool of choices to fight with in NV than in skyrim. In Skyrim you're limited to melee, bows, and magic. And there's really not much you can do with all of those things. While NV you can set traps with mines against enemies that would otherwise roll you. NV is far less "Mindless". You don't get to just use 1,000,000 health potions, since stimpaks are generally rare. You've got to worry about conserving ammo.
Graphics, Skyrim. NV used the same graphics that were used in FO3, a 2008 game. Skyrim came out more recently and the graphics are beautiful in my opinion. If you can max them out. Do it, if you can use HD texture packs, hell yes. This game is so Beautiful. You won't like NV's graphics unless you love dirt.
The progression in Skyrim, and NV are so different I find them hard to compare. In new vegas, you better know what you're doing or you'll find yourself at level 30 without the Attributes, Skills, Or perks that you want. While In skyrim you simply max-out everything eventually, becoming god-mode even on the harder difficulties. In fallout, It's less about your experience and more about your gear. While in skyrim if you've got 100 one handed you can fuck shit up with a wooden sword. Both are awesome in my opinion. But I lean towards Skyrim because in NV(without mods of course) the game sort of ends once you find the best gear and get to level 30. Not much you really can do past that except run quests that at that point are generally-piss easy. While reaching that point in skyrim would take you HUNDREDS OF HOURS. Unless of course you're a dirty booster. The smithing system can easily be abused giving you the option to make godlike-gear at around level 20. It's best to avoid this in my opinion just so that you're not "Godmode" and focus more on quest-related gear which is generally cooler anyways and spend your perks on combat or other crafting systems like alchemy and enchanting. No points for either here though, since I feel that I am biased otherwise, Skyrim all the way.
Quests and story are the only thing I don't really have an opinion on. They both aren't really that intense and are somewhat generic. Which in my opinion is a good thing. In open-world games the best things are generally the sidequests. Id the main story were VERY compelling you may feel like not doing the sidequests. Like I do in games like mass effect and dragon age. Both games have factions you can join, all with their own line of interesting quests. That may be tricky to find your way into without a google search like the brotherhood of steel, and the dark brotherhood. I think that NV executes this better because your relations with other groups is affected by your current affiliations.
Lore type stuff. I know that this should be considered in immersion I think that it's a completely different thing altogether. It's something that's always there but not necessarily noticed if you don't pay much attention. I give skyrim the advantage because It has 9 different races all with their own backgrounds. Several decades worth of lore and a much larger world. Filled with magic and shit. But lore is for nerds. So whatever.
Damn this was long and boring.
anyways, TL;DR: Both games are worth playing and one should not be ignored for the other.
EDIT:Shit, look at that wall of text
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