The idea is that the game code is running on the blockchain itself (not the rendering to cute kitties though), so there is no game server that can be shut down by the developers. So even if the game stops being profitable and the devs lose interest, the game could presumably keep going and everybody would still have access to their kitties and trade them.

(except in this case there is a self destruct function that would freeze the game forever)

The benefit is that it's a tradeable asset / collectible - that is, you can buy and sell it. Use it to mortgage a house, get credit, etc. It's like (real) money that's the difference. So far $20 million traded. On the blockchain you have transactions recorded (not in a private database but public and shared who owns what or who own how much etc.). Cheers. Prost.

I want to be able to trade kitties with people without Kitty Bank of America having to manage and facilitate the transaction, for one.

Guaranteed scarcity for one.