Sorry, lets try this again.
I considered the new Tomb Raider a huge achievement.
I always envied the PS3 crowd for, what I felt was a better Tomb Raider, the Uncharted series. The levels were beautifully designed, the story lines compelling and well-written and the gameplay was fun, various and had a surprising amount of depth to it (I consider the second as the best of the Uncharted series).
Tomb Raider gave me all of the above, and so much more.
I really enjoyed the fact that this game really emphasizes exploration in the sense that it encourages you to find all the things (even allowing you to backtrack to previously visited areas). In my opinion, this adds well to the survivor/archaeologist image of Lara. It also adds quite a bit of play time to an already quite long game (of course, incomparable with the TR games of yore).
The combat in Tomb Raider is different, but it's where the franchise needed to go. The first Tomb Raider revolutionized 3rd person gaming... but then failed to improve on that achievement. If you consider other 3rd-person-action games, they all have some sort of free-aim shooting mechanic (Max Payne, Uncharted, etc), it makes sense for Tomb Raider to have the same. And the shooting in Tomb Raider is top-notch. The salvage mechanic means that you grow to really love your weapons, and adding a little bit of stealth means that outsmarting goons is incredibly satisfying. This is not a dumb shooter. In fact the AI scripting felt incredibly atmospheric, especially when enemies react to your actions ("she's hiding behind that pillar, flush her out!" - "damn she's a good shot!" - "where did she get that grenade launcher?"). Seeing the enemy react to the growth of your character is also immensely gratifying.
Story-wise, this game is very different. Gone is Lady Croft, and in her place is a younger, not quite there yet Lara, put in a situation that would push most people beyond their edge. I must say I do not agree with what you said about the Tomb Raiding, most of the older Tomb Raiders were not set in literal tombs to begin with (the only true Tomb Raider in that regard is The Last Revelation) and Lara visits quite a few Tombs in this game (especially if you consider that the entire island is a tomb, what with all the corpses lying around). But the game is not about Tomb Raiding, I'll give you that, it is about Laras literal ascent from the darkness, which also reverbs in some of the gameplay mechanics (such as gradually leveling up and getting stronger). I loved how different the approach to the story was, whereas Uncharted is obviously influenced by flicks like Indiana Jones, Tomb Raider seems to take its inspiration from the survival horror genre (indeed, Tomb Raider especially appears to be channelling the Descent, many scenes in the game make explicit references to the series). I loved watching Lara get more and more confident about herself and her strengths.
One of my favourite scenes: Picking up the grenade launcher
All in all the story feels a lot more cohesive than any Tomb Raider that came before and also than the Uncharted games, as it feels so much more grounded.
Like I said, I was waiting a long time for a Tomb Raider that would match Uncharted 2. What I got in return was a much more personal experience that rocked Uncharted clear out the water. I loved playing this game so much, that it is the first game in which I collected 100 % of all the artifacts, challenges, maps, documents, etc... And I did all that twice! My only regret is not playing this game in hard difficulty first, not because it was particularly easy, but because I know my experience of the literal ascension would have been an even greater one. I love this game and will most definitely return.
edit: why the rants about missing puzzles? There are quite a few in this game (you obviously haven't been in the tombs) and imho they make a lot more sense than the old tr puzzles.