Firstly let me say that there are no active, magnetic pickups. If we look at say EMGs, they are (or were) low impedance pickups that had a pre-amp or booster encapsulated within the pickup housing. This kept the noise down as high impedance pickups can be noisy due due to their higher output. It had some advantages and some disadvantages. In the early days, putting some of the electronics on the pickup meant a smaller control cavity and lower noise However as components have become smaller and we understood how to shield and wire more effectively these advantages have all but dissipated. It also meant that each pickup needed at least 3 wires instead of the two on a passive single coil pickup.
Low impedance coilshave a lower resonant peak that is often outside the range of the instrument meaning a more balanced output.
These advantages of "active" pickups were, in my opinion, fewer on bass than on guitar so you will here less difference between them.