I can't think of many other games where the mechanics themselves reinforce the story. Papers, Please does it pretty well.
Also consider playing the following if you have thirst for puzzle games
- The Swapper
- Snakebird
- Spacechem
- Starseed Pilgrim (this one I haven't played but Blow recommended it, iirc)Also check out The Talos Principle [0], it resembles The Witness (based on little clips I've seen of it), and feels like Portal with a dash of Isaac Asimov and a sprinkle of Religion.
Beautiful audio, beautiful scenery, clever puzzles. One of the best games I've ever played.
I've replayed Limbo, but never Braid.
FWIW I've revisited Braid multiple times, but don't think I'd get anything new out of Limbo if I played through it again.
I don't think I'd consider them better games than Braid, however, they just have (much!) more replayability/content.
(I would consider them better than Limbo, but that's neither here nor there).