Sinply:
Passive crossover - most speakers- the low and high frequencies are separated by a network of inductors and coils, ensuring the frequencies go to the correct driver (woofer low f and tweeters high f). Inductors filter out high f but they're big clumsy imprecise things that reduce the amplifiers' control over bass leaving it wooly - you hear a drone and fluff, instead of attack and timbre. The capacitors are relatively high value, so their accuracy is also not optimal (unless you measure and match pairs ... which costs).
Active crossover- using precise resistors and tiny value capacitors with op amps the same job is done with much more precision. This time the crossover is placed between the pre-amp and 2 power amps. There is nothing between the amp and speakers, meaning they exert full control over the driver ... when a drum kick is hit, the speaker starts moving and stops moving exactly when it should - it doesn't carry on, so there us no smearing of the bass.
Other things come into play- especially with passives, like phase differences that can completely ruin the stereo imagery.