A BB squat holds the bar higher up on the traps (usually with the hands placed closer in on the bar), thereby allowing a more upright torso position, thereby throwing more of the stress on the quads. A powerlifting squat involves holding the bar with a wider grip further down the back, which allows you to lean over further, involving more of the glutes and (to a certain extent) hams. Since the bar is closer to the lever point of your hips, you can generate more power this way.
Personally, I think that if you’re after huge quads, you can probably use a board or weights or whatever under your heels and it won’t hurt you much. People with good ankle flexibility sometimes use a board anyway (even though they don’t strictly need it) to throw more emphasis on the quads if they’re doing a bb movement. Sort of like with a front squat.
Also, you’ll find that whether or not you need a board may vary with how wide your stance is. I have very poor left ankle flexibility due to an injury, so I always use a board when doing shoulder-width (or narrower) squats, but with a sumo stance I find I can do without. You can also experiment with the degree of foot turnout, as different foot positions will yield different stresses.
How this helps. BTW, you posted on another thread that you’d gotten some good gains out of your HST cycle. Care to post some stats?
cd: Hate to say it, but I’m out of tag lines. Since 2002.