The reason why your full-width-div doesn't stretch 100% to your screen it's because of its parent "container" which occupies only about 80% of the screen.
If you want to make it stretch 100% to the screen either you make the "full-width-div" position fixed or use the "container-fluid" class instead of "container".
see Bootstrap 3 docs: http://getbootstrap.com/css/#grid