There are more types of pickups in the world than just 2.
Sure you can generally divide most pickups into 3 basic pickup designs: single, humbuckers or splitcoils (which technically are humbuckers functioning as singlecoils), and there will be general inherent tonal qualities associated to those 3 basic types of pickups.
However if there were only 3 different tones to get from pickups there most properly wouldn't be room for as many pickup manufactures as is the case, as well as there wouldn't be any whatsoever reason to use different pickups let alone swap out the ones your bass would already be equipped with.
I am not exactly any expert on pickups, but explaining which pickups has which tonal qualities would be a subject for a rather comprehensive book rather than a post on a forum anyway.
To sum up the 3 basic qualities inherent with the 3 base forms of pickups typically used in an electric bass though:
Singlecoil: Usually have a more clean, clear and articulated reproduction of the string vibrations, but also have a relatively lower output than your typical humbucker and can be perceived weak or thin in comparison.
Humbucker: Have a stronger, fatter and heavier reproduction of the string vibrations but as a trade off is not as detailed and defined as your typical singlecoils.
Splitcoil: Usually used as a way to get rid of the inherent hum issue associated with singlecoils run on their own, as a tradeoff for the lack of noise, while it is possible to gain a higher output level than your typical singlecoil pickup, it usually isn't quite as articulated as if it had been a true singlecoil.
As I said I am no expert, so please someone correct me if I got something wrong in my description.