The main advantage of leaf springs is that they perform the functions of the elastic element and guide device simultaneously; hence the low price of this component. Truth to say, it is also a kind of a drawback - as a spring has to bear several types of loads at once: pushing force, the vertical reaction and torque reaction of the bridge. As a result, leaf springs are less reliable and less durable than the spring suspension.
The main problem of the springs is that it is very difficult to make them soft enough. The softer they are, the longer they need to be manufactured that leads to the fact that these elements become prone to the S-shaped bend. Besides, springs have friction between the leaves, moreover this friction may be unpredictable. It depends on the condition of the leaves surface of the sheets. Read about elastic elements of the suspension.