Dust caps are for keeping (as the name implies) dust out of the "engine" of the speaker driver. The diagram below shows a cutaway view.



If enough dust or debris gets into the area between the voice coil and the pole piece, you'll have friction in a place you definitely don't want it. Dust cap-less speakers, like those found in many subwoofers, simply have the voice coil attached directly to the underside of a bowl-shaped driver.