The brightness up/down buttons are there to adjust how bright/dim you want your screen, but even on its dimmest setting, the screen isn’t completely off.
To turn off your screen, here are 2 options I like…
1. use a keyboard shortcut: control + shift + eject
2. set up hot corners on your screen: apple (in the upper left) > system preferences > expose & spaces > active screen corners > (set a corner to…) “Put Display To Sleep”. I’ve got mine set to put the screen to sleep if i drag my mouse to the lower right hand corner of the screen