A year ago I went to a large conference on school improvement (the focus was on using test results and assessment to determine instructional priorities, action plans, etc.). There were a number of presentations from schools that had made significant progress. My principal and I went and chose to attend presentations on low-income, high-diversity urban schools, since we figured that would be most relevant to our situation. We both noticed that nearly all of the presentations (teacher and principal teams presented what they had done and details of their implementations, etc.) were from Catholic schools.
What stood out was not so much the specifics of what they did, but the commitment of the faculty. They were clearly dedicated and willing to go more than the extra mile in a way that would be unusual in a public school, even one with very competent and caring staff. Whether it was their faith per se, or more of an ethos of service, it was quite strikingly different from what one would encounter in a public school, even a good one (and I have worked in a few).
Now of course these were "success stories," and it does not follow from this that every Catholic school is similarly graced with highly committed staff, but the factor of shared vision is probably an important one that should not be underestimated, and is difficult to replicate in a diverse public system, where many values are simply not shared.