With regard to power and clarity, very simplistically:

Let’s say that a certain amplifier has a maximum output of 20 volts before clipping sets in and provides a gain of 20. Let’s also say that the input to the amplifier is receiving an input signal of two sine waves mixed together, 0.7 volt of 200Hz and 0.3 volt of 3200Hz, and this feeds a two-way loudspeaker that has a passive crossover at 800Hz. With a gain of 20, the amplifier will provide 14 volts of 200Hz and 6 volts of 3200Hz on the same output, adding to 20 volts at some points of the resulting waveform, which will be split between the woofer and tweeter respectively.

With passive bi-amplification, the same scenario exists except that two channels are outputting the 200/3200Hz mix. One output goes to a crossover which blocks the 3200Hz and lets the 14 volts of 200Hz pass – this output channel has almost no load at 3200Hz but still must use 6 of the available 20 volts reproducing the 3200Hz. The other amplifier output goes to a crossover which blocks the 200Hz and lets the 6 volts of 3200Hz pass – this output channel has almost no load at 200Hz but still must use 14 of the available 20 volts reproducing the 200Hz. So each amplifier channel will have a reduced load, but does not provide more voltage at each individual frequency. This load splitting may be helpful in some ways, or hurtful in others, but no significant increase in power is gained for either frequency, and passive crossover elements are still between the amplifier and the drivers.

With an active crossover, the channel handling the 200Hz signal does not have to handle the 3200Hz signal. The 200Hz channel can now receive a 1-volt 200Hz signal and output 20 volts at 200Hz. Increasing the 200Hz voltage output from 14 to 20 volts doubles the power available at 200Hz. For the 3200Hz signal, increasing the voltage by the same ratio, an input of 0.43 volts instead of 0.3 results in an output 8.6 volts instead of 6 volts.

If the wattage delivered in a passive crossover scenario, with either single or bi-amplification, were 33 watts to the woofer and 6 watts to the tweeter, with an active crossover and increased amplifier input voltages, the power delivered can now be 66 watts and 12 watts respectively.

In the real world, one would not have two clean sine waves mixed together. There would be a mashing of many, many frequencies with narrow transient spikes all over the place. In the two passive crossover examples above, if high frequency transients were added to the mix, either they would be lost because the amplifier has no headroom left, or the overall input level would have to be reduced so that the peak voltage output required, transients included, would never exceed 20 volts. In the active crossover example above, the low frequency channel wouldn’t have to deal with the spikes at all, and the high frequency channel still has 11.4 volts of headroom for transient spikes even when the low frequency channel has reached its output limits.

Without a lot of further details, one could conclude that as far as maintaining clarity while raising power output is concerned, in very rough terms, two 50-watt channels fed by an active crossover are as just as useful as 150/200-watt channel(s) using single or bi-amplification into passive crossovers. By extension, two 200-watt channels fed by an active crossover are just as useful as 600/800-watt channel(s) feeding passive crossover networks.