The advantage that maverickronin mentioned is a substantial advantage:
...Separating the crossover from interdependance with the LCR components of the drivers makes tweaking and adjustment easier.
This is the reason that plenty of people (Siegfried Linkwitz for one notable example) are (was) averse to building speakers any way except with a dedicated amp for each individual driver, with the amp placed between the crossover filter and the driver. Designing a passive crossover filter is almost a fool's errand because the impedance of the driver is so poorly behaved. You can use a Zobel to offset the rise in impedance that is due to the coil inductance, but this won't do anything about the impedance peak associated with the driver's resonance. Owing to the typical sharpness (high Q) of the impedance peak at the resonance, it poses a very difficult problem for a passive network between the amplifier and any driver other than the woofer. For the woofer it is deemed unimportant so long as the output impedance of the amplifier is as low as it generally is with solid state amplifiers. For drivers other than the woofer, the difficulty posed by this impedance peak is due to the fact that at the frequency where it typically occurs, the impedance of the high-pass filter is in roughly the same ballpark as the impedance peak, such that the voltage split between the two impedances is very different from the intent for the high-pass filter. To mitigate this effect (but not suppress it entirely), common practice is to use a pair of resistors, one in parallel with the driver to flatten the driver's impedance, and the other one in series with that parallel arrangement, to keep the impedance of the parallel arrangement from being too low. This is a useful technique but does not fully eliminate the problem. In practice the pair of resistors is deemed adequate, however if anyone were inclined to fully suppress the impedance peak such that the high-pass filter could fully behave the way it is supposed to, nearly a dozen passive crossover components would be needed just for this particular purpose, which is more than the total number of crossover components for both filters, typically found in a two-way speaker. This is more economically feasible when a buffer (amplifier) is placed between the filter network and the driver, but then it becomes moot, because when the high-pass filter isn't between the amplifier and the driver, the impedance peak is no longer the same problem.
It is much, much easier to make the filter do what you want it to do, when the amplifier is between the filter and the driver rather than the filter being between the amplifier and the driver. But this doesn't squarely address the question that was asked, which asked more specifically about the use of active filters. There are major advantages of putting the filter in front of the amplifier, as I explained above, and there are further advantages of using an active filter constructed partly of operational amplifiers, as opposed to a filter constructed purely of passive components. The advantage of the active approach occurs when the filter needs to be a high order filter that requires multiple stages. When multiple stages are needed, the law of diminishing returns comes into play. As the number of stages increases, the incremental benefit of adding another stage approaches insignificance. To get around this effect, operational amplifiers are used as buffers between the stages, which isolates the stages and allows each stage to be as effective as it would be if it were the only stage. Additionally, the use of op-amps accommodates series filter topology, where the balance of impedance between two filters in series determines the filtering effect of each. In active filter design the series approach is often taken for granted, and the principles are fundamentally different from the principles of parallel filters as are generally used in passive crossovers.