When you have two distinct things, which you understand to different extents, proving that they're identical allows you to learn about one thing from the knowledge of the other thing.
To use your example:
Prior to today, we knew that cats lapped up milk with their tongues, and also preen their fur with their tongues. Also prior to today, we knew that a Japanese animal called neko coughs up balls of stuff.
Today we found out that cats are identical to neko.
We now know that cats cough up balls of stuff, and that neko preen their fur with their tongue. We might now use this new-found knowledge to identify that the coughed up balls of stuff were probably derived from fur and/or milk.