From what I understand private prosecution is not cheap or easy.
Also private prosecution would allow for a citizen to charge someone with a law violation that a prosecutor would never charge anyone with. I would give some examples however, due to the fact that most Americans refuse to believe the truth I will not provide them here.
Maybe you work at a place that is committing the crime of conversion against you. You have taken all the steps you can before making it a legal issue. You file a complaint for felony conversion and conspiracy to commit conversion with the prosecutor's office. The prosecutor wants the business/company to continue to commit this crime. With private prosecution you still have to prove your case and you're the one paying the costs (I hope you get reimbursed if you win). Then you can still file the civil charges.
I will try this as another example. Taking a motor vehicle without permission (theft of an automobile). Let's say that an ex BF/GF has a copy of your car key still. They are still upset about the break up, take your automobile, and crash it into a UPS truck repeatedly (this really happened). If the prosecutor looks at it and says "I am too busy and it's civil offense," you can still charge them with the felony (under Washington State law) that it is. Maybe the EX is a friend of the (public) prosecutor and that might have some affect on the (public) prosecutor's willingness to prosecute.