I was staying out of it, but I think the wood, the pickups, the strings, the amp, the speakers and the player are what determines the sound. You change any one of these and the tone changes.
In general though, the acoustic sound of the guitar is critical. If it is dead acoustically, then nothing will revive it - even BK's!
I have many guitars which are similar, but even deciding which pickups to put on a guitar sometimes is a task. I have had more than one set on about half my electrics, and have kept changing them sometimes (even BK's).
Elliot, did you ever try the pickups the other way around (JV Apaches, MIM Mothers Milks)? It would be interesting to see if that makes a difference. Also what is the acoustic difference between the two?
Wood types make a huge difference too, but it is hard to generalize because each piece is so different. Wez had some mahogany for a neck, and two pieces which were bought at the same time from the same place wieghed totally different amounts (one felt like twice the other).
All of it makes a difference. That's why I like to hear a guitar acoustically before I buy. When I bought my Goldtop, I played it acoustically for about 10 minutes and then said I would buy it. The guy was shocked that I hadn't plugged it in, but as I explained to him, it had Tom Holmes pickups (which I had never used), he didn't have an amp like I used in the shop, and I could change the pickups and eeletrics anyway if I didn't like the sound. Acoustically it sounded great so I was happy to buy it. So the wood has a big part in the sound.
But all the rest are factors.