BBF is great, not because it is more accurate, but because it is more flexible and can allow you to operate the camera more quickly/conveniently without having to switch between AF modes. Two main advantages....
1) You can leave the camera in AI Servo (Continuous) AF mode and you can simulate One Shot (Single) mode by briefly pressing the BB to acquire focus and then releasing it;
2) You can take a picture without having the camera (re)focus when you don't want it to, without having to switch in and out of MF all the time.
As a combination this is great because you can track moving subjects easily, and if/when they stop moving it is no problem to be able to focus and recompose by simply releasing the button. You can then release the shutter as often as you like without having the camera refocus somewhere else. Equally you might be shooting a static subject, such as a bird in a tree, where you might use the centre point to focus, and then recompose, but when the bird takes flight you instantly have AI Servo ready at your thumb tip to start tracking the moving target.
Nothing at all to do with improved accuracy. Just potentially more convenient, but it depends how/what you choose to shoot.
EDIT : Third advantage - if an obstacle momentarily gets (partially) in the way of your subject you can temporarily stop focusing but continue shooting, resuming AF once the obstacle has passed.
The thing is, focusing is one operation whilst taking the shot is another, so why have both operations tied to the same button? Surely it is better to be able to choose when to focus (and when not to) and when to take the shot.