Looking through my current mod selection, the non-graphical ones I consider vital (since I have a tonne of "holy shit hi-res mods". I have 3GB of VRAM and I aim to use it all!):
Realistic Battle And Dynamic Combat:
http://www.newvegasnexus.com/downloads/file.php?id=34921Now, I'm the first to cry "Too FPS" with some mods. I think New Vegas should be predominantly an RPG, so stats should play a vital role (i.e if you have a low guns skill you shouldn't be able to hit shit). That being said, this rebalances weapon DT/DRs to something sensible, so if someone is wearing Power Armour a BB-Gun is not going to bring them down just because you're damned persistent. It basically makes the way the numbers get crunched feel more Fallout 1/2.
FO1TAG:
http://newvegasnexus.com/downloads/file.php?id=35347Restores the Fallout 1/2 style of Tagging skills, where they rise by 2 points instead of 1 when you level them up. It just feels better and makes tagged vs non-tagged skills still relevant after five levels.
Logical Reputation Shifts:
http://www.newvegasnexus.com/downloads/file.php?id=34944Makes it so if you shoot one dude to death that faction doesn't vilify you for life. Now you have to be a bit more of a dick to make everybody hate you.
Project Nevada:
http://www.newvegasnexus.com/downloads/file.php?id=40040Worth it just for the sprinting and bullet-time alone, and the toggleable stealth boys are neat too. Since sprinting and bullet time eat up your action points like delicious chocolate I consider this not-too-FPS.
Radio New Vegas: The Secret Stash:
http://www.newvegasnexus.com/downloads/file.php?id=37029Adds 80 more songs to the radio, all of which keep the amazing 50s theme. I love old jazz
WME - Weapon Mod Expansion:
http://www.newvegasnexus.com/downloads/file.php?id=37576There are a lot of competing mods for Weapon Mod tweaks, this is the one I tend towards.
Also:
Missing "Inventory is a Backpack" from Oblivion, I made a custom mod that drastically lowers the amount you can carry (from 150 + Strength*10) to just Strength*15, and then added a rucksack that you have to drop and open but can carry infinite weight (largely because I haven't figured out how to limit the amount you can put in it). It basically forces you to think ahead as to what you're gonna take out of your inventory before-hand, since you can't access it in combat. When I can figure out things like not making it a bag of holding I may release it...but there may be a mod out there that already does this that I missed.