The real reason for the disparity (well, most of it) is that charter schools can kick out kids who either don't behave, have special needs, or are simply low-performing. I teach public school in L.A., and we see it every year in the spring in California when the CST is taken. The charter schools kick out their worst kids just before the test to raise their scores, and the children show up in our admissions office about a week before the test.
The two biggest problems that I see are 1) we cannot kick kids out of middle school. It doesn't matter if they start fights, set fires (yes, really), vandalize property, curse at teachers, bully smaller kids, have weapons, sell drugs, disrupt the classroom every single day... we cannot kick them out. At best we can find another school to transfer them to, if that school will take them. That school will usually offer to swap one of their own little sociopaths and we agree, hoping that if we peel a kid away from his little crew he runs around with, the shock might make him straighten up a little. Of course, we can only do this with the parent's permission. (I say parent because there's usually only one.)
The other problem, I admit, is the union. We can't get rid of bad teachers either, but I have to say (though no one on FR will believe me) that bad teachers aren't as prevalent as people think. Most people who can't teach realize it and quit after a year or two. But a few do hang in there, and they can be a bit of a bummer. But even a mediocre teacher can do pretty well if the children are well-behaved. They just can't deal with the wild little thugs we get. I've been doing it for 10 years now and let me tell you, it's like juggling on a tightrope.
OK, so let's do the math. You take 1,000 kids from a low performing student body that has 5% of the kids pass a state exam. In order to make those 50 kids (5% of 1,000) account for all of the rise to 88% passing rate in the charter school, you would have to kick out 946 of the 1,000 (50/56 = 89%). That does not account for 824 of the 880 (or 93%) who pass the exam in the charter school.