Okay, so I got the new Tomb Raider as part of AMDs current promotion (get a 7850 or 7870, you get TR and Bioshock Infinite, get a 7950/7970, get Bioshock Infinite and Crysis 3). I am currently playing it, and will be updating this on my progress.
The first few minutes are right out of the Uncharted playbook (and going by the reviews, things will stay uncharted for some time), something I really like. When I had a PS3, the Uncharted titles were some of the highlights, having a similar experience on the PC is something I wanted for a long time.
Now, the story begins with Croft being an archeology student on an expedition to find relics of some ancient asian queen, hidden in what the game calls "The dragon's triangle", a Bermuda Triangle with the serial numbers filed off. There, the ship they're travelling on gets caught in a storm, breaks apart, and sinks (as you would imagine). Lara and other survivors wash up on the shore of an island, and are promptly kidnapped by a cult that likes to string people up and cocoon them in sailcloth.
EDIT:
Okay, so I'm past the first major combat sequences. Comparing this to Underworld, there's a lot more characterization here, the stakes are a little more believable, the characters are good so far. The gameplay also flows nicely (Caveat: I am playing this using a 360 pad, results with KB/Mouse may vary), the gun/bowplay has a nice weight to it, and the areas I've been in have all been very rewarding in terms of exploration. There isn't much of it, and so far there has been a distinct lack of big open vistas that Underworld would throw at you again and again, but we'll see if that comes up later.
So far, the game does a very good job of reminding you that Lara is just one character on an island where everything (except maybe the chickens) can and will try to kill you, and leaves it up to you to figure out how to survive. Verdict so far: Pretty good.