Basically, a passive in-the-speaker crossover network has inductors in addition to capacitors, where a low level active network only has capacitors. Inductors in series with a woofer introduce resistance from the wire - it can't be eliminated, and this effects damping. There is interaction between the passive filter sections for in-speaker crossovers - this is supposedly the reason for "passive bi-amping" to eliminate this interaction.

An active crossover can have much more complex slopes with more precision than a passive network. The capacitors can be very high quality polystyrene or polypropolene with close tolerances. They are also a much lower value of capacitance.

A specific amplifier can be used for each function, such as a tube for the highs and solid state for the lows.

If one amplifier clips, the other one is not effected by that clipping, and keeps playing cleanly.

It is easy with a variable crossover to change the crossover frequency.

I'm sure there are advantages I've missed, but these are what comes to mind now.