Let's start with the title which doesn't make any sense.
Charters Offer More Choices in Harlem, but Stir Concern for Public Schools
Charter schools are public schools. What the Times means to say is that charters are stirring concern for non-charter public schools (or whatever retronym you prefer).
Even the definition given by the Times is off.
Charter schools, which are publicly financed but have latitude in how they operate
What the Times means is that charters aren't constrained by the legacy of conditions and agreements they agreed to or had imposed upon them during the decades of non-competition which, while generally true, misses the main distinction.
The main difference between charter schools and non-charter schools is that charter schools have to earn their customers, i.e., students and then retain them, thus, creating a quasi-market.
This is, of course, no guarantee that charter schools will perform any better than non-charter schools. Many won't. The market doesn't guarantee winners. The market works by ruthlessly weeding out the losers.
It's pretty clear that the Times doesn't now why or how a market works. But, they are quickly learning that lesson as they slowly drive themselves out of business as their financial losses continue to mount.