An active crossover is implemented before the power amplifier so you need to have an amplifier for each driver. For this reason it is also referred to as Bi-Amping, Tri-Amping etc.
A speaker with a passive crossover can present a complex and difficult load to the amplifier, taking into account the impedances of the drivers along with the inductance and capacitance in the crossover network. In most cases active will be a significant improvement. There's a good article here that describes in much more detail.
Another advantage of active is that you don't need the enormous (and expensive) components required for passive. I'd suggest seeing if someone has already designed an active crossover for your speakers.
Just to be clear I'm talking about active analogue crossovers (using op-amps), not DSP based.