I'm not particularly technical, but I'll try to give the explanation I have experienced. In some subs (like my JL F110 Fathom) there is only a low pass filter to roll off the upper frequency range of the sub, meaning you run your main speakers full-range and have to adjust the sub to integrate with them. If your speakers go down fairly low in frequency response, this might be difficult to do without having some overlap in frequencies which also excites some room nodes. I have been wrestling with this problem, as my speakers go down into the mid 30Hz range, and the lowest crossover point on my sub is 30Hz; I cannot get rid of a room node at about 50 Hz, even with the sub's equalization program, because the main speakers excite the room node a little by themselves, and even with a steep rolloff the sub's high frequency response accentuates that bump. .An outboard crossover can take the full range signal from your preamp and split it at the chosen crossover frequency above the low frequency cutoff of your main speakers, so that your main speakers get a signal that is rolled off in the bass--this way they don't have to run full range and can have a crossover point that can be adjusted for room effects and better integration with the subwoofer. Some speakers also may benefit from having not having to reproduce lower bass frequencies, at least from some of the reviews I have seen.
As a potential drawback, there is the additional circuitry and interconnects, but a high quality unit like the Bryston is likely to add very little, if any, signature to the sound. Also, in subs which have a high pass filter for the sub and a low pass filter for the main speakers, an outboard high quality active crossover might be better than the built-in crossovers in the sub, and result in better sound. So yes, there is more flexibility, but you can play with crossover points and slopes and potentially get better sound as well.
Hope this makes sense. Perhaps others with more technical knowledge can chime in.