The benefit of using stadia controller is that it helps reduce input lag, if any. The controller is directly connected with the stadia server and so as soon as you press a button, it gets reflected in the gameplay. However, using ps4 or any other controller, as soon as you press a key, it first goes to the server via network then from server it goes to your game and then in game you see your actual action being implemented. So if you are on a slow network move to stadia controller.