Disclaimer: don't mind the amount of hours this review shows. I'd played through this game on a physical non-Steam version back when it came out.

Max Payne and Max Payne 2 are among the best video games ever made and are in my personal top-10. And this is the case when a sequel made about half a decade later by another developer is almost as good as the originals. Yes, many disagree, but this is a near-perfect sequel, and here's why:
1. The gameplay conveys the spirit of the originals instead of trying to copy them, hence feeling both innovative and nostalgic at the same time. Every element of the game-design is made to remind you of the past without feeling repetitive. The only completely new feature is cover-shooting, but it's integrated in such a natural way. You never just sit in cover and shoot everyone like in other games with this mechanic. You're still constantly moving because your enemies are constantly trying to flank you or destroy your cover, so in essence the gameplay involves more tactics as you move from spot to spot, barely keeping yourself alive, gobbling up painkillers. It's intense and addictive. This game stands on its own, it's not just a clone of the previous games with a few new features. And yet never for a moment you will feel like it's not Max Payne.
2. The story is emotional and suspensful, carrying over that cynicism and wit from the previous games, yet taking itself more seriously this time. The creators knew they couldn't just make it a simple continuation of the story. It would've never lived up to its predecessors. So they've put their own spin on it. This feels more like an epilogue than a direct sequel, in a similar fashion to The Godfather Part 3. Yes, the story in the previous installment was self-sufficient, yes this third entry didn't need to exist, but it does, so they take this one in a different direction and explore a new side to the main character.
3. Neo-noir. This point is more of a continuation of the previous one, but I felt like this needed to be addressed separately. Ever since the game came out I kept hearing people complain: "Max Payne 3 is not noir". It is. And if you disagree, you don't understand what noir is. While the original 2 games paid tribute to the 1940s' black-&-white movies, this game references the neo-noir of the 60s and 70s, which had a tendency towards sunny, hot environments, social alienation of the protagonist (i.e. being lost in a country where nobody speaks your language), exploration of societal issues such as poverty, prostitution, drugs, etc., the existence of a darker underbelly (i.e. organ trade), etc. All of these elements are present in Max Payne 3. Not only that, you can clearly see the noir structure in plot: things constantly get worse and worse as you progress. You complete an action-packed mission, getting closer and closer to your goal, but in the end you fail. Every step forward seems to be dragging you further down to hell with bodies piling up and things spiraling out of control. How is this not noir? This type of structure is one of the core elements of noir.
4. The physics and controls work in perfect unison to convey the sense of age of the protagonist and immerse you into the game. Max doesn't always immediately react to your inputs, but it's done in such a natural way that you never feel like the game is broken, but rather it sells you that illusion of being older and weaker. And yet throughout all this, you aim better than ever. It's almost like Max knew he was getting older and started to rely more on accuracy than agility, understanding he's not as fast anymore. The element of cover-shooting fits perfectly into this theme.
5. The soundtrack. I don't have much to say about it, aside from it's amazing and fits the game's art direction, gameplay and story perfectly. It's not the same music from the first two games, and yet the main theme from them plays at important moments, reminding you yet again: This is Max Payne, but it's also its own game.

Conclusion: Every step of this game is designed with an incredible amount of care and attention to detail. This game is an experience, where every element of it fits perfectly with the rest, and where the whole is better than the sum of its parts. I honestly can't imagine how else this game could be better. If I really HAD to nitpick, the only thing I dislike about it is how Max keeps switching weapons before every cutscene. I think it was probably done because they could only record cutscenes once and had to pick which weapon they'd want to feature in them, but at the same time they wanted to keep continuity and fluid transition between cutscenes and gameplay. I personally would prefer that they break this continuity in favour of gameplay, but still this is such a minor issue. I think, were the first 2 games not better in regards to the story and atmosphere, I would've probably call this the best game in the series. It's such a surprise to see a different developer nail the sequel so well. This is a 10/10 type of game. A must-play. A masterpiece.