This is my take.
The full squat requires much more hamstring flexibility than the low bar squat because you're squatting so much lower.
Now what are the hip flexors responsible for in the squat? As Steven said, the hip flexors help to create an anterior pelvic tilt. If you don't engage your hip flexors, it's much easier for your lower back to round, which means the posterior pelvic tilt or butt wink.
Because with the full squat it is so much harder to not round your back, to counteract the pull of your glutes and hamstrings, your hip flexors and abs have to fire that much more.
Hamstrings + Glutes --> posterior tilt & butt wink
Hip Flexors + Abs --> anterior tilt