A core is just a fancy name for an emulator, like an “app” or “application” is for “program”. And a save state is a full dump of a given program’s memory and that can be reloaded later. A game save is, to my knowledge, a checklist for the game to load onto memory.
Save states are good if you can’t rely on game saves, like if your device has low battery and you’re far from any save spots, if you’re in the middle of a very hard section, etc.
Meanwhile, as memory is physically located in a given device, it can be found in a different place if you use another update of the program, another installation, another OS, and perhaps even another hardware. And if a piece of memory isn’t where the program expects it to, the program won’t load at best.
Also, though rarer nowadays, some older games had bonuses if you had the game saves, the sole save format back then, of others (usually previous) games from the franchise. Naruto Ultimate Ninja 5 and Dragon Ball Budokai Tenkaichi 2, for example and if memory doesn’t fail me, gave you money if they detected saves from, respectively, Ultimate Ninja 4 and Budokai Tenkaichi, while Persona 3 FES allows you to carry over the compendium of Persona 3 saves and Final Fantasy X can bring over from other saves of the same game items needed to understand the language of a group in the story.