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.
What Am I On About Now?
The reason this matters is because I frequently see people mistake one for the other when it comes to diagnosing a problem or prescribing a solution. If you had a poor experience with a game then you need to ask yourself if the problem was the system or the content. Fun needs to come from somewhere, but there is a fine line you'll have to tread between the procedure providing the fun and the adventure providing it.
For example, I've met plenty of people who don't think the D&D combat system is fun. And while I am not saying that this is an incorrect opinion to hold (there are many reasons you might be justified in feeling this way!), I am saying that every single one of these people, when asked to go into detail, actually described a problem of level design instead of game design. The D&D combat system can be extremely fun... if the DM has arranged a scenario which brings the fun out of it. If you just have the party fight a handful of mooks in a blank, flat, open area, then yes, it'll be boring. Instead, you need to take advantage of the environment and use chokepoints and elevation and cover and then have secondary goals during the combat and 3rd parties to protect and constraints on what can be done and enemies with special abilities that change up the situation and so on.
Ideally, good game design should be able to stand on its own (like controlling Mario). But that's an ideal, not reality. And to be honest, how long could you spend chasing down that rabbit in Super Mario 64 before you'd get bored? So in reality, you need good and interesting and fresh content and you need to keep providing more of it regularly. So the real advice to follow is this: the best rules in the world won't do shit for you if you don't have good content to use them with.
The system should provide for both of these things to some extent. Most games will try to provide 100% of the game design you'll need and might provide a little bit of level design if they're feeling nice. Personally, I'd like if there was a bit more balance. I don't mind a "rulings over rules" philosophy and I like how 5E D&D is so modular and DIY-friendly. Meanwhile, I think it should become a much more common practice to always provide a sample adventure or scenario to use your rules with when you release a new system. To me, the biggest obstacle to trying out different systems isn't having to learn their rules, or even to teach those rules to my players. It's having a good adventure to use them with. Making well-designed adventures is by far the most time-consuming part of DMing for me. My standards are high and I oftentimes have to make up almost everything myself because I'm not being offered anything to work with.
Luckily it's been a tradition in D&D since the beginning to offer an entire manual full of monsters to use, so having a combat-centric game means you'll be pretty well covered for level design concerns. But when it comes to other activities, there tends to be gaps.
A major, glaring one is the scarcity of good wilderness adventure content in mainstream gaming these days. Focus moved away from it in 3E D&D and it's never come back. It's one of the most frequently criticized parts of 5E D&D. I'm planning on making a ridiculously long series of posts about it at some point. They're in my drafts folder slowly growing, don't worry. But I'm gunna jump the gun a bit and talk about how this concept of Game Design vs Level Design is relevant.
Think about the balance. Some people like simple rules that don't get in the way, and instead want to focus all their energy on making good-ass content to play in. Other people really like systems and challenge that's derived from the rules themselves and how they apply. I would say that an RPG with really good game design can be considered a "strong" system when it's doing a lot to create interesting gameplay just by merely using the rules in their most basic form. A good magic system is usually interesting and challenging to use on its own. A good slot-based encumbrance system can always provide a little bit of extra challenge to deal with when used no matter what's happening in the adventure. A kingdom-management game can often be quite self-sustaining in its challenge just with the built-in limitations of how little the players can accomplish vs how much they need to accomplish to stay afloat.
So the main takeaway for game design is that you shouldn't consider a design problem to be "solved" simply when you've had one good session of gameplay taking place in that space. Because lots of folks will "solve" a problem like "wilderness adventure" or "urban adventure" or what have you just by creating a list of scenes they'll throw at the players specific to their current adventure, and that's it. But that's not an actual, sustainable "solution" to a design problem. You solve these things when you can make a reliable set of rules or procedures that'll always work for playing any session with content in that space. If you made an animation of Mario jumping around and you had the player tap the button alongside it like they're playing Dance Dance Revolution then you haven't really created a platformer. You solve the platformer problem by figuring out how Mario always moves, regardless of the context.
Meanwhile, the main takeaway for level design is that you'll always need some amount of it, no matter how strong the system is. There are lots of folks thinking about game design all the time. But there aren't as many folks who'll put that effort towards the accompanying level design we need in order to appreciate their game.
You can make a great open world campaign that'll fall flat if you haven't seeded the world with interesting places to find and interesting things to happen to the players.
You can make a great detective/mystery procedure to use that'll fall flat if the mystery itself is boring and the clues aren't well thought out.
You can make a great naval combat minigame for your pirate campaign that won't save it from inevitable doom if you don't have tons of piratey things to do and piratey islands to visit and piratey NPCs to interact with and piratey problems to reckon with.
Go forth with this insight and hopefully improve whatever it is you're working on or thinking about.