Public Radio- The main advantages to having public radio stations in the United States (or anywhere) is that, by removing their need to make a profit, you remove their need to compete with other radio stations for actual ratings. You can hire professional journalists, play extended clips and interviews, include specialty shows, and limit advertising. These things tend to limit bias in broadcast much more effectively than in private radio. It gives listeners an alternative to mainstream media.
The main disadvantage is that it requires taxpayer support as well as donations to stay on the air, so funding is continually shaky. some would argue that public radio cannot as easily cover world events and reports without using the same wire services and media outlets as traditional media.
Private Radio - As with all private companies, these stations deliver what the people want to hear, both in information and programming. Those stations that do not do this lose ratings and go out of business, so you could argue private radio is more responsive to the public.
The same argument is also a disadvantage. By catering to public whims, they are more likely to try and entertain rather than inform. So we have a lot of one sided talk shows and narrow programming of music per channel so that the listeners get what they want, as well as the advertisers who influence content.