With Braid, you learn actual things. And you can tell this because you can immediately re-solve puzzles you've solved before. Contrast this with Stephen's Sausage Roll or Jelly no Puzzle -- these have a way more "combinatorial" feel to them, such that you can "re-enjoy" the game more, but they are also less interesting to me on that account (Replay value correlates inversely with learning actual things) -- a game should teach you about its world, in a way that creates lasting insight.
I will call this the "actually learning actual things" test. Math is an actual thing, but many courses don't actually teach it, so you don't find it any easier the second time around. On the other hand, you might try hard and actually learn something, but if it's like Stephen's Sausage Roll, you won't get better at it by much. (There will be some speed up because you will know the "rough shape" of the answer from declarative memory, even if your instincts aren't any better.) Jelly no Puzzle is sometimes an intermediate case, where there's a "big insight" followed by lots of low-level jelly-pushing; in that case, you'll remember the big insight but not the low-level mechanics.