Its little old post, still I’m adding more details incase in future if someone comes here.
Game state is like a object state that exists on server and replicated (all) clients.
game instance (GI), in analogy you can consider it as running exe process. Which exists on both client and server but they are independent and there is no logical relation between server GI and client GI.
For example when you start Fortnight game, that is nothing but Game instance, which is used to switch between widgets, querying master server and Join game session (match), switching between game modes (quad PvP, duo PvP or PvE). In all the cases you have only one game instance exists. It Gets destroyed when you close the game exe.
But GameState will be recreated for each game mode (session/match). When you play quad PvP you get one game mode, after that match finishes, if you join another PvP, you will get new GS.
Hope this helps. Sorry for typos.