Bruce Baker of Rutgers (and schoolfinance101) has been pointing out for years that all charter successes are completely explicable by looking at two factors: (a) increased spending and in-school time (as at HCZ etc.); (b) different populations. If you spend more money and time on a school, and if your student population has fewer poor, ELL or special ed students, then your test scores will be higher. Baker has repeatedly shown that highly touted charters are actually spending more money and/or enrolling more advantaged populations.
The New Orleans schools have both increased spending (as the NYT article mentions but passes over quickly) and a distinctly different population (as the article does not mention at all, as far as I could see.) This second point is important. The population of New Orleans is distinctly different post-Katrina (as the NYT reported). It is smaller overall, it is more white and less black, and it is less poor.
My guess is that these factors are easily enough to explain the higher test scores.