This is how you do a remake. Anniversary stays true to the spirit of the original Tomb Raider and just tries to improve things, rather than throwing everything out and making a new "modern" interpretation. This game is especially enjoyable for a pleb like me who finds the idea of the oldest TR games appealing, but finds the tank controls much too unintuitive to enjoy.
It's a very atmospheric game focused on exploration, that gives you a strong feeling of isolation. I loved this, and it's a stark contrast to Legend, which tried to emulate explosive Hollywood blockbusters and had Zip and Alistair constantly chatting your ear off. Level design is one of the game's strengths, and it manages to be sprawling and complicated without being confusing or labyrinthine. The gameplay here is primarily about platforming and puzzle solving (usually both of those things combined together). In my eyes, the puzzles are usually the perfect difficulty, being challenging but not obtuse, and rewarding once you figure them out. Traversing the environment feels good most of the time, but shows a lack of polish at other times; like when you miss ledges because you didn't jump at exactly the angle the designers expected, or when you're wrestling with the camera to try to get it to look in a useful direction. The camera is one of the bigger problems with this game, and in awkward positions it can be difficult to assess your movements properly. Sometimes you just have to take a chance that might result in a long fall to your death.
Combat isn't much different from Legend, except for being more subdued, which really helps. I liked how you're only fighting animals and supernatural creatures here instead of squadrons of soldiers. There's now an "Adrenaline Dodge" move, where enemies become "enraged" and charge at you, at which point you can dive to the side and headshot them in slow motion. I don't mind this, but the fact that they become invincible while they charge at you makes it feel more gimmicky and can hurt the flow of combat. Enemies are still bulletsponges in general. The game is peppered with quick time events just like Legend was too, but they're easier here, and mainly just used to make the boss fights a little more interesting, so I actually think they were used effectively (for the most part). The boss fights are more elaborate than they are in Legend, and less repetitive. A smaller criticism I have is how I need to constantly mash the right trigger on my controller in order to keep shooting the pistols, instead of just holding it down. Sounds like a nitpick, but it's uncomfortable.
I wish they made more Tomb Raider games like this one.