As briefly as possible:
- active preamps allow you to boost and sometimes cut bass/mids/treble (or a combination). Like active tone controls on 90% of modern bass amps. They also buffer the signal, which makes the bass sound more "independent" from what you plug it into (cables, pedals, amps...), with more consistent sound. For this reason, they may also sound a bit brighter even when flat (depending on implementation). Active preamps may also boost volume, but they are usually designed to be unity gain.
- active pickups are wound with many fewer turns of wire, which, for various reasons, makes the pickups sound much more extended in the treble than passive pickups. They need to be active (so with an internal preamp) because otherwise the signal level would be too low for most amps, and anyway, they benefit from having a preamp right next to the coil (less noise).
- conversely, passive pickups usually sound darker and with more colouration in the upper mids due to the presence of a resonant peak. Active pickups sound brighter and "flatter", passive pickups have that classic character.
- with active pickups and/or an active preamp, you can plug into a passive DI, whereas passive pickups need an active DI (otherwise they sound dull).