The benefit is to do with autofocus - when you look through the viewfinder what you are looking through is effectively your lens wide open, regardless of what aperture you are shooting at.
For example, say we want to shoot at f/5.6 and you have the choice of a 300mm lens with a max aperture of f/2.8 and one with a max aperture of f/5.6, in theory both should perform the task equally well, right?
If you mount each lens on an identical camera and look through the viewfinder though you will notice a big difference, the image from the 2.8 lens will be much brighter, despite the lenses both being set to the same aperture.
The reason for this is that your lens is wide open until you press the shutter button, only then does it close down to your chosen aperture. The 2.8 lens therefore allows much more light through, which in turn makes it much easier to autofocus. In practice what this means is you will be able to nail the focus much easier in lower light with the 2.8 lens than with the 5.6.
My own lens progression went like this, 70-300 VR, to an 80-200 f/2.8 and on to my current 70-200 VR and believe me, the difference between the 70-300 and 80-200 is night and day. I'm not saying the 70-300 is a bad lens, far from it, however the 80-200 and 70-200, being "pro" glass are simply in a different class.