Bullhorn bars are traditionally used in the Pursuit and Time Trial events where the extra control that drops afford isn't necessary. (You're really not ripping on the bike in a Pursuit like you are in a sprint)
However, bullhorns put the rider in a far more aero position and, as we all know, aero is good.
"Quäl dich, du Sau!" (trans.: "Suffer, you swine!") - Udo Bölts
bullhorns and aerobars are used for non-mass-start events. mass start events are where competitors all start at the same time, in a mass - drop bars are requires for mass start events. non-mass-start events are time trials - an individual or a small team against the clock. some are run as "pursuits," where two opponents start at a time trial at opposite sides of the track, and if one is caught by the other, that person is eliminated.
aerobars get a cyclist's body into a position resembling a downhill skier's aerodynamic tuck, and are part of an aerodynamic repertoire that usually includes an aerodynamic helmet (think The Rocketeer) and wheelset (discs or with thick carbon spokes).
I'm not sure what you mean by "teeny weeny bullhorns."