The fact that I can build upon CryptoKitties with no obstacle. If CryptoKitties was on a private database I would need to get access to their API and also trust them with the ownership/management of the assets. Now I can just build it on the blockchain and have people destroy CryptoKitties in my smart contract to get benefits in my game. No API functionality or permission from CryptoKitties needed.

So how does this open API works? Every time you use it you have to pay a third party?

You submit the action you want to do, anywhere in the ethereum network, and if it's valid (e.g. if you're running an operation that destroys a kitty then you'd better include a digital signature showing that you own that kitty) then it will be accepted and published. You have to pay a "gas fee" which in theory a) should be pretty small and b) is what pays for the running of that ethereum network. In practice under the current implementation those fees got to be big enough that they kind of killed cryptokitties, but this might be fixable.

What do you mean? All of the data is on the chain. It's a global "third-party" that everyone is a part of.

Even "the ability to monetize" is hardly unique - it would also be trivial to monetize that database table (see: Diablo 3).

That the two projects both shared the same DB.