When planning a trip I've seen people say private cars are scams and other people say they can be better than city taxi's because you know how much it will cost ahead of time. Are there any other advantages that a private car may have over a city taxi? Are there any other disadvantages other than the cost? Do city taxis have hidden costs that a traveler should be aware of (such as "running the meter" while waiting for a rider)?
When using a regular taxi make sure it's with a reputable company or you agree upon the rate upfront. In a lot of countries taxis will take every opportunity to rip off tourists.
You'll generally not have this problem with private car services.
Private car services are also generally more comfortable and easier to have little stops or detours along the way.
This completely depends on the city and situation. I've only used a private car service once, in Lima, Peru, as the taxi drivers are notorious for ripoffs and theft. When I arrived at the hostel I was staying at, there were two people staying there who had all of their belongings stolen by their taxi driver at knifepoint. An Australian tourist was also recently shot by her taxi driver in Lima after refusing to hand over her belongings.
In retrospect it seems like that was $20USD well spent, but I wouldn't bother paying it in places that didn't have a reputation.
Taxis aren't run by the government everywhere. For example, here in the Czech Republic the government recognizes two types: taxi and yellow taxi (the main difference being that yellow ones can use taxi stands and special lanes) - but both are private cars by your definition.
It is not possible to answer the question in a way that applies globally because there are different conditions in different countries.
My experience is that a normal taxi is always cheaper. The only advantage of a private car service is that the cost is fixed unlike a taxi where the price on the meter (if there is one) can vary depending on the route, how long it takes or how much the driver feels like charging. There is also the possibility of getting ripped off by a taxi driver. Also, the private car services tend to have newer cars and in my experience the drivers can be a little less "crazy". Finally, I've found that in some cities that the only taxis that are possible to call by phone are private taxis and that regular taxis can only be hailed on the street.
I travelled to Peru with an organised Tour. We had a guide who accompanied us for the duration of the Tour. On the final day, when I wanted to get a cab from Lima to the airport, I could have paid an official taxi for about $30. However, my guide knew someone who can drive us to the airport for about $10 to $15 cheaper.
If someone you trust is providing a private car, you can feel safe to take that instead. It really depends on your comfort level before you accept a private car from someone.
In some countries, there are regulated and non-regulated 'taxi' services. It so happen that those are generally in countries that are less secure. In those place, absolutely take the regulated ones!