Application settings on macOS are ususally saved somewhere in ~/Library: Common places are ~/Library/Preferences/com.example.mycoolgame.plist for preferences (should be in plist format and "com.example.mycoolgame" should be a valid bundle ID that you own (you should own the domain)). The advantage/disadvantage of this path is that power users know about this directory and can edit the files there as they wish.
Then you have ~/Library/Caches for cached data. All cached data should be put somewhere under this directory. (Never use it for content that cannot be regenerated or redownloaded though.)
~/Library/ApplicationSupport/YourApplicationName/...: here you can basically do anything you like. It would be good if "YourApplicationName" would be globally unique in this case.. So better make it long. Users usually don't see the filesystem contents of anything below "~/Library", so there is no need for short names.
Of course, you can also put your savegames in ~/Documents/MyCoolGame/savegames and tell the user that you save the games there.
A gotcha (maybe): I'm not sure if system APIs expand "~" properly. I think probably not: Calling fopen with a path that starts with "~" would most likely not do the right thing. The users directory is located at something like "/Users/max", so "~" expands to "/Users/max" in the command line if the username is "max".
I can't answer your original question "What directories does Linux have that macOS doesn't?" because I don't have a Linux box at hand at the moment, and I don't think that it would be helpful for your use case.
For global data, there is also the "/Library" hierarchy.. But normal users don't have access to this place, so your game would need to ask for admin rights, which will make everything much more complicated, and this will feel user-unfriendly to macOS users. The macOS way is to have a self-contained application bundle and put all user-specific or temporary data into the appropriate place within the users home folder.
