The single most compelling reason FOR ME to change was the realisation that back button focussing allows for TWO modes of focussing without having to change any settings.
1) AF-C: continuous focussing - just hold the back button down whilst you follow the action. (this is what I would do for 97.256% of my shooting
2) AF-S: single shot focussing - pick the subject element you want to focus on. Press the back button until it acquires focus, then RELEASE ... recompose to suit.
Until I went to back button focussing, I could not do BOTH without having to change the focus mode setting - often inconvenient or missed the shot.
That has been the ONLY advantage for ME in what I shoot - but it is a big enough advantage for me to stay with it. It is taking some time to develop the coordination to control two buttons in a shooting sequence rather than just the shutter release.
I like BBF because it allows you to actually shoot in
three modes at once:
1) AF-S
2) AF-C
3) Manual
If you only ever feel a need for AF-S I don't think BBF is all that useful, though I still prefer it to the delicacy required to half depress the shutter. When it's below freezing outside, it's way easier to just press a button to focus, then let go instead of half depressing a button.
A great example of where BBF is basically a necessity is a football kickoff. You want the focus to be squarely on the ball, so that you can get the kicker striking the ball at moment of impact perfectly in focus. But you'll also need AF-C for getting the kick returner catching the ball and making his return. If you shot the kickoff in AF-C without back button, then the camera would try to refocus on one of the other players on the kickoff team. If you shoot in AF-S with half shutter pressing to lock, then A) you have to half depress the shutter for a long time, as it's difficult to know exactly when they'll start the kickoff, and you want your framing setup before that point. B) you'll then have to quickly change focus modes while the ball is in the air, find the kick returner in that, then hit him with AF-C. BBF you just compose your kickoff shot, get focus locked on the ball and let go of the button. Then after the strike is made on the ball, you simply find the returner and depress the focus button again.
Further, if you want to be in manual focus, there's no need to flip a lever, you simply just don't depress the button. Very useful when you have all sorts of things around your subject, but your subject is relatively static. Even with AF-S, in that scenario as soon as you take one shot, the camera might try to refocus on a closer object on your next.