Because with a belt pulley, when you change directions the tension remains constant and there is no "play" providing a more consistent positioning of the pulley. With a gear there is "play" or slack in the gears to allow them to roll freely. But when you change directions, your positioning will not be as precise. If gears were installed without play or a small amount of slack they would wear out prematurely. Belts can be installed with a tight - somewhat stretched fit and will provide much more accurate positioning while transferring power.
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It would depend on what you want to use the belt and pulley for
crowned pulley keeps a belt tracking on it
No it is not. A belt, for example, goes around a pulley.
A car engine
Snub pulleys are used to increase the angle of wrap of the driver pulley or and to change the direction of belt(deflector pulley)