I believe covert acts like sneak of open and closing doors and containers. It does not prevent you from coming out of hiding. It believe it prevent anyone in the room from seeing you do it.

Improved hide lets you do all that (plus tells and cb) from the shadows. So similar, but presumably (and I can't verify) you'd see the door open and close, but not my thief stepping out if it was just improved hide. With both, you see nothing.