Game design is when you make rules and procedures. It's answering the "how" in how things work. It's the description of how skill checks work, or how combat works.
Level design is when you make content with which to use those rules. It's answering the "what" in what the players are doing. It's the adventure module that tells you which skill checks to roll, and the encounters of monsters and battlefields where combat will be happening.
When Super Mario 64 came out in 1996, it was a smash hit and a breakthrough in gaming. It was the perfect 3D game, seamlessly translating the 2D genre of platforming into a 3D context better than any other attempt to do so. And trust me, the other attempts failed hard. It was an exceptionally tricky and ambitious design goal to tackle, but once they got it right, it blew the doors wide open for the future of 3D gaming. And you know how they did it?
First they designed the mechanics for Mario's movement. That's it. That's the only thing they focused on initially. They created the little minigame of chasing down the rabbit and catching it, so they'd have a way of testing their system. But they worked their asses off to make sure, above all else, that it was fun and easy to control Mario. That merely having to run around and jump on stuff and use your different moves was strong enough on its own. Only after they nailed that down did they begin to design the courses that would be in the final game.
First they nailed game design. Then, when it was so good it could be fun just by itself, then they poured their hearts and souls into making incredible levels. But the point is that these are two separate steps, and two separate goals. So I want to talk about the role that each one plays in tabletop design.