So far it seems like she runs a very authoritarian, orderly society in her Ice Kingdom, as opposed to the Gum King's kingdom which is just wacky, zany fun.
Ice Queen seems to be super strict and no fun while Gum King is essentially the Ice King from the main show, but more jovial.
Ice Queen is yandere for Gum King.
As for her origin, maybe she's just someone from before the Mushroom War who was super into science and tried a Cryogenics thing to survive. And she just ended up making herself accidentally immortal with Ice Powers, so she gathered up other ice-creatures in the colder regions and made a kingdom out of it.