So what can't I do when I go to other EU country that you can when changing state in US? I still get health care, I still can work there, I can move there if i so desire. What can't I do what I could possibly still want?

You are a foreigner when you move to a different place across EU, with all social implications. A worker moving from LA to NYC is not nearly as disadvantaged as one moving from say Sicily to Sweden.

Of course you are able to move freely and there's no legal barrier preventing your employment, but in practice you are rarely on equal footing with locals. There are both linguistic and cultural externalities at work here, which are simply absent over the pond.