The bib martingale is normally only found at the racetrack, and as X says above, it makes for young horses who may grab at the martingale with their mouth being unable to get caught up like that (which results in a wreck). It also is heavier on the reins, which, if you have the lighter feel that you would normally have with a riding horse instead of a racehorse, can be a bit of a problem. The bib martingale also works as an Irish martingale, holds the reins from being separated, which may or may not be useful. I have both in my barn, after several decades of riding and training racehorses as well as a longer history in the h/j show ring, and lots of green horses through the decades. I don’t use the bib martingales on show horses, there is no need for them there. For a riding horse, I don’t want that much weight hanging on the light feel I want on the reins. For a racehorse, which classically takes a bigger hold on the reins, the added weight of the bib doesn’t matter.
PS. neither “prevent head tossing”. The use of a running martingale only keeps bit pressure on the bars of the mouth, no matter what the horse does with his head. If a horse tosses his head around, without a running martingale of some sort in place, the angle of the pressure changes from the bars onto the corners of the mouth. This, in itself, can cause stress and misunderstanding in the horse, and some horses simply don’t respond well to pressure being elsewhere other than on the bars, and the change in the angle of pressure is in itself, a stressor. The running martingale keeps the pressure off the corners of the mouth, even if the horse raises his head, thus is a calming influence on a horse like this. If you wish to use pressure on the corners of the mouth, you remove the running martingale.