The default mode is "Auto". In this mode, when the camera detects you putting your eye up to the EVF (viewfinder) it automatically turns off the LCD and turn on the EVF. Take your eye away and it turns the EVF off and LCD on. You can also set up the camera via the menus to have only the LCD OR only the EVF turned on. You can't turn them both off at the same time, nor I think, would you want to since it would make it impossible to use the camera.
Fuji is a great camera but very expensive. The one thing that turned me off about Fuji was their slow autofocus. Maybe they have worked that out. Besides that the sony and Fuji are pretty close in terms of performance. It is really up to you and how much money you are willing to spend.