Charter schools are funded by the government and must follow policies set by the government. While some charters have apparently successfully positioned themselves as "private schools for free", they are not. Many in the media have fallen in with this and frequently contrast charter schools to public schools which is a false dichotomy. Charter schools are public schools and should more accurately be referred to as "public charter" (as opposed to "public district").
Why do they do this and why does it matter?
Firstly, the essential distinction between a public district school and a public charter school is that the former is generally funded via and answerable to a district, and then perhaps a county, and then to the state and then the feds while the latter funded via and answerable to the state directly and then to the federal government.
Public district schools have a significant extra level of administration, bureaucracy and politics. For political reasons, state governments specifically remove many of the controls placed on public district schools so as to encourage people to set up, run and attend public charter schools. However, the states can just as easily put them back.
For example, public charters must follow testing mandates, ELL mandates, curricular mandates. Requiring all teachers to be certified has been long-rumored. Private schools must do none of these as they exist in a true free market - tell people what you offer and if they don't like it, they won't come.
Secondly, all public schools are funded according to political priorities. Should the state decide all children must learn Greek, or must play water-polo and the funding will follow. Alternatively, should a public district or a public charter school decide, for example, to limit classes to 15 students, it will have to find the funds from elsewhere in the budget if the state's staffing budget is for 20 / 30 / 40 students per teacher.
Non-public schools do not have these constraints. If they decide to do or not to do something, they must pay for it themselves through fees, fundraising or financing. If people do not like it, they won't come.
So why does it matter?
Public schools depend on the political winds of the day, and not on student / parent satisfaction. A change in government or a change in policy inevitably means a top-down non-educator-driven change in public schools, sometimes overnight. Non-public schools depend on delivering what has been promised, and thus on student / parent satisfaction.
Parents choosing schools should evaluate their priorities and then determine which of the public-district, public-charter or non-public schools best matches. However remember, there is a continuum of accountability and while a public charter school may be more accountable to you as a parent than a public district school, politicians and politics still matter more than you do.