I guess my question assumes I have two suitable power amplifiers which can drive the speaker within spec in either configuration.
What audible differences would I notice between the two configurations (bridged vs bi-amped)? I realize the bridged speaker could play louder, but at lower settings and same speaker sound levels, would there be an audible acoustic improvement in the bi-amped configuration?
I tried to explain this earlier, but the two situations you are asking about are apples and oranges.
Let's pretend you have a stereo amplifier that has the ability to be put in bridge mode.
Right now you have two speakers; a left and a right. Each speaker has its own channel. Now you bridge the amp and you now have two speakers connected to only one channel that has the sum of power from left and right channels, but only one output. In other words, instead of stereo, you have monoral.
Will that sound better? Probably not. So why do it? Answer: you shouldn't.
Let's talk about bi-amping. Bi-amping requires four amplifiers for stereo sound. Two channels are assigned to the left and right woofers and two other channels are assigned to the left and right tweeters.
Why would anyone do this? Well, it is not because you get more power. You do this because you eliminate some nasty side effects that would normally be present from passive crossovers. The primary problem with crossovers is that you get phase delays at the crossover frequencies that muck up the sound a bit.
When you bi-amp your system you use an electronic (active) crossover that allows you to adjust the exact crossover point and you get very little phase delay. As a bonus many electronic crossovers allow you to "time" align the tweeter to the woofer.
Time alignment is also a phase issue present with many speaker systems. At and near the crossover point the same frequency sounds will be produced by both the woofer and the tweeter. If the two drivers are physically at different distances from the listener (usually the woofer is slightly further than the tweeter from your ear) the note that arrives to you will have a distortion created by the two speakers being slightly out of phase. This causes a distortion in the note's waveform because of the lack of time alignment. Think of it as two ripples on a pond that are made when two stones are dropped simultaneously. If one stone is closer to you than the other, the closer stone's wave reaches you first, but the second stone will be close behind. As the ripple mix they will either reinforce or cancel each other out to some degree.
The disadvantage of bi-amping is mainly cost because you need more amps and an electronic crossover. You have to either have speakers that are made to be bi-amped or you must modify them (remove the crossover and add a second set of speaker terminals). And there is a lot of work to set up the system to make it play correctly. Bi-amping is not plug and play. You need to set the crossover frequency, set the gain for each speaker's driver, and set the time alignment for each driver, too.
The advantage will be better sound in the end, but getting there is not a trivial task.
If this doesn't help or make sense, then I would recommend going to a library and reading anything you can get your hands on about the subject in general.