Okay, this is all going to be a little simplistic, but hopefully usefull.
(By the way, Thaiman, just log out now, this is a biggy
I'll only consider the tweeter for now, as the woofer is basically the same but the other way around for frequency.
The tweeter has a high-pass filter between it and the amplifier that stops any low frequency signal going to it. If it didn't it would zing merrily past your ear the first time somebody hit a bass drum, probably before. What this filter does, regardless of whether or not you're bi-amped, is ramp up the impedance that the amplifier sees for low frequencies to that driver. With one amplifier driving the whole speaker, the amplifier still sees a normal impedance lower down because of the low frequency part of the speaker. In a bi-amped set up it doesn't. The impedance for high frequencies hasn't changed, but that for low frequencies has.
So what does this do for the power output?
First a question - what does an amplifier actually do? - it provides voltage gain - i.e. for a given input voltage the output voltage is multiplied up by some figure, limited by a maximum output voltage which is the output from thepower supply. This hasn't changed at all - the amplifier is still increasing the voltage by exactly the same ratio it was before, but the impedance it sees has changed for the low part of the frequency range - it's gotten much much bigger. So the output power, which is basically dependent upon the voltage and the impedance of the load is the same for the treble part and much lower for the bass part. The other amplifier has the opposite effect. Overall you do add the two outputs together, but they're not the same outputs you see on the spec sheet anymore because of the changed impedance that the amplifiers see. And the net result is more or less the same as before you started.
At least this is true for a passively bi-amped system. For an active system, you remove the crossover from the speaker altogether and put it in place between the pre-amp and the power-amp. So each amplifier sees a different input signal - a lower input voltage because it's for only part of the whole signal. In this case the headroom for each amplifier is increased because it doesn't see a large part of the signal and the output is ... well ... bigger. It's not doubled (not in the real world) because the bass signal is probably the higher amplitude part and that will clip well before the treble does, but you'll definitely get more by doing this.
In a passive system you will actually get more too, but for a different reason and to a much lesser extent - amplifiers have two limiting factors - the maximum voltage they can supply and ditto for current (a bit more woolly this though). When the power supply in the amplifier is trying to provide too much current the voltage that it puts out will droop slightly, more so the more current is being drawn - this equates, with non-regulated or poorly regulated power supplies (and plausibly with regulated ones though only if the voltage drops quite a lot) to non-linear amplifier behaviour since the output is essentially the input * max output / max-input and max output is the output of the power supply. So it will "clip", but not in the hard-clipping sense of the input voltage exceeding the maximum allowed that you would see with an infinite load impedance. I've referred to this a "current-clipping" before and been totally slated for it, but I don't know of a proper term. Possibly "gain-clipping" might be a better term? I don't know and would welcome advice if available.
The above isn't very clear is it?
Basically - bi-amped the amplifier is being asked to provide less current, which makes it's life easier and so it can work better. You might, at the limit, get more power out because of this. (How often is the amplifier at its limit?)
But going active is the proper way to do it.
Oh, and in the real world lots of more modest speakers don't actually provide a low-pass filter for the bass unit, they just use the natural electrical and acoustic properties of the driver to do the job for them and just put a capacitor across the tweeter. It's not really an issue, but thought I'd mention it anyway
I hope this helps at least a little, and if you think it's rubbish feel free to provide a better explanation