I need to start with the the bad first, because while not changing the fact that I definitely recommend this game, I feel it really needs to be stated.
The bad aspects of this game really come down to one thing: it makes too many assumptions on what you are trying to do VS what you should do.
Let me explain.
You can say what you want about the original, requiring quite a bit of precision with its blocky walls and ledges, as most of the moving around was made by jumping and grabbing ledges that were always facing each other. But it was clear when and where to jump to and from. The game knew what it did, and didn't make any assumptions of any kind on what you were trying to do. When you died, it was entirely your fault and your fault only.
The Anniversary edition has gone beyond that and allows you to grab onto a ledge while jumping from another perpendicular ledge. While this is welcome and freshens up the game feel most of the time, it gets tiring when the game does not agree on your trajectory and lets Lara fall off to her doom without having her trying to grab said ledge. Next time, you try an identical jump and surprisingly the game now agrees to it.
This is but an example. Overall, what happens is that the game does not allow you to do things it is not expecting.
The most infuriating aspect of this is when the game decides wrongfully what is allowed and what isn't.
You can reach that pole from where you stand? Lara won't grab it because you are expected to pull that cage over there to be able to reach that ledge over there that will in turn allow you to grab the pole.
This leads to some really cringe-worthy moments were you will fail sequentially until the game finally decides to let you do what you actually are trying to do.
Finally, and this is more of a detail, but still thought I'd mention it.
Remember that animation in the original, when Lara lands on her feet and hands when she falls off a high platform? That is an animation that tells you "this is as high I can fall from" and you're like, "phew, that was close". You could really feel that fall through Lara.
In Anniversary, Lara falls off, lands on her two feet like nothing happened physically, still yells and loses some health, but that animation is not there anymore. Tough falls do not feel like tough falls (except those you die in). Too bad.
All of this being said, this remake of the first Tomb Raider is everything a remake ought to be: it freshens up the original with new ideas and does not try too hard to be like its predecessor.
New graphics (2007-wise, but still look good in early 2015) and environments. If the sensation of vastness was there in the original, Anniversary builds on top of that and adds a more detailed architecture that improves that sensation of vastness.
New controls that use the gamepad perfectly (don't know about keyboard, playing with a xbox 360 controller).
New combat system that works way better than the original's jumping around. You still jump around here, it is just more dynamic overall. Even though combat is not frequent, there are enough to break the slow going pace of the puzzle solving. Also, boss fights! That means you have to find that specific thing to do to beat them.
New cutscenes, cinematics and new voice acting. Some cutscenes have QTEs, but this remains very rare and anecdotal and does not add much to the game except giving you the impression of control when Lara does some badass stuff on screen.
Collectibles that will have you rummage through the levels if you want to get all of them. These unlock rewards such as new costumes, artworks, developer commentaries, etc. There is also a time trial mode in which you must complete a level in one go, without checkpoints or anything: you fail, you restart entirely.
Some of the old puzzles have been replaced by new ones, but old ones that worked well have been kept. In the original I sometimes found myself wandering aimlessly until I eventually came across all the pieces needed to solve a puzzle. I never felt that in Anniversary. It was almost always a straightforward but still rewarding experience.
In addition to the ability to jump to perpendicular ledge, there are other new gameplay elements such as a grappling hook, swinging poles and climbable poles, all of which open up new possibilities for exploration and puzzle solving.
I played Anniversary along with the original to fully appreciate the differences, and in my opinion, Anniversary is a great reverence to the original. It stands out on its own, with fresh ideas that definitely make sense for a game like this, while not casting any shadow over the original which has its own merits over Anniversary.