Would it be possible for someone to record a mono vs stereo clip so I can hear the difference?
It's possible (although I really don't have the time or inclination myself) but if it's just clean signals, you're really not going to notice much.
The real magic happens when the 2 signals are treated and effected separately and in different way which you can really (or easily) do with a single signal.
For example, Billy himself keeps the low end quite tame, some moderate compression, and the SVT amp is clean, and maybe a little eq rolling of the top end slightly. The P pickup goes through a pre amp for grit (which also has a blendable second distortion channel, in effect giving 3 signals!), bags of compression, into another SVT with the low end almost completely rolled off. He also has an Eventide effect unit in this path for the odd bit of harmony in a couple of solos.
The beauty of this is you have complete tonal control over both pickups separately, and eq'd the way he does, the high and low end don't fight for sonic space, ands stops it getting too muddy. Thus it allows him to get a really compressed top end to bring out harmonics and tapping, but the gently compressed low end still has room to breath and vibrate properly. The other bonus is he can get overdrive and distortion going which are low end suckers, but running the other signal clean blends them together.
You could achieve similar with a multiband compressor, and some clever processing, but splitting at source just gives better control, and a greater degree of separation.
I have seen him play with his bass set to mono through one amp, and he does still sound like Billy, but you could tell the high and low end were fighting for sonic space.
There's a section on his Advanced Bass dvd that goes through his entire rig that is well worth looking into. It may even be on youtube by now.
In fact my band is doing our demo soon, I plan to go direct with mine into 2 channels, one for each pickup, exactly for the above reasons. I think the engineer may even love me for it! Kinda like DI'ing and micing an amp at the same time.
Hope this all helps make things clearer.