"NOT necessarily differences in how they approach or think about the game."
I'd say it's a little bit of both. I think about the game very differently than when I was an 1800 for example. For example I think a lot more about how choosing to close or open the position can have a big impact on how easy the position is to play for me; I think a lot more about trading the right minor pieces; I'm much more enthusiastic about outplaying my opponent by setting up a good endgame for me, rather than being afraid that too many trades equals a draw. I'm a lot more practical. All sorts of things.
I think that also depends on experience. I've played a few ~10 year olds rated 1900 and I'm not sure they understand much of anything except how to calculate and a pretty good amount of opening theory.
I have to admit some of the tactics they can find in seconds are really impressive though.
Maybe there is something unique about prodigies like that. There is only so much wisdom you can have then, so maybe early on they have to replace that with unbelievable tactical ability. And then as they become IMs and GMs they start to get that wisdom. Me getting to 2000 was absolutely nothing like that, but then I'm not a prodigy and never was.
Over the board ~ 1950 FIDE blundered horrible checkmate in one vs 200 points lower rated opponent with his king on middle of the board with just few pieces left with a lot time left on his clock. He even asked not to write down last move... I would not say 1600 is the average rating where opponents stop blundering on simple tactics or checkmate threats. Tactics matters and players lose games because they did not saw or calculated bad a basic tactical combination and thats pretty common in sub 2000. Some players rated 1700 are great at tactics and would not drop/miss tactics, but many do. Some play very solid openings and never blunder vs equal rated opponents.