A passive pickup is a high-impedance coil wound with a relatively high number of turns - commonly over 5000 turns on a single coil. Part of the high impedance is the high resistance - these coils are typically between 5kΩ and 12kΩ (with the "fattest" pickups being in the 30kΩ range - the Gibson sidewinders, for instance).
An active pickup is a low-impedance coil wound with a small turn count - commonly under 2500. The resistance of these pickups is usually fairly low, on the order of 1 kΩ. In order to bring the pickup output to a level similar to that of passive pickups, active pickups have a built-in preamplifier that runs on a 9V battery - and this one actually works as an amplifier, not just an active set of controls.
The good thing about active pickups is that the low turn count means less hum picked up, and, if made with a good preamp, even single-coils are commonly quieter. However, the downside is that active pick-ups simply do not work without the battery (the output is negligible), whereas with a passive bass with active circuitry you can always bypass the circuitry and go fully passive if the battery goes flat.
As far as tone goes - I wouldn't hazard to guess. All my basses have passive pickups - of those, two have active electronics, but as you've already said yourself, it's not the same.