ADS-B vs ADS-C
When they say peer to peer it means that the 2 sides (the airborne station and the ground station) perform a "hand shake" after the airborne element has logged on to the system That's part of the ADS-C system. the C stands for contract, meaning both have entered into an agreement to exchange certain information and instructions.
This is normally in the form of some type of subscription agreement between the aircraft operator and the service provider. It's more detailed than that but that's about a simple of an explanation in layman's term needed to understand the concept. CPDLCC could come under this agreed link in certain scenarios, but it does not have to be part of the ADS-C agreement and although they are tied together, they are not the same. ADS-B on the other hand does not require a subscription agreement.
If an aircraft is equipped, and turned on, the aircraft determines it's position continuously and will BROADCAST an updated info package every second. There is a lot more in the message than just position, but that's beyond the scope of your question. Another significant difference between the 2 is the timing, or cycle of information sent, received or exchanged. As stated, ADS-B make a broadcast out every 2nd with specific data. With ADS-C, it is around 20 minutes for a 'normal' report. That could change depending on the need of the ground station, or if the pilot enters an emergency mode, or certain other events aboard the aircraft trigger an immediate update transmission. ADS-B has the accuracy, reliability and reliability to be pretty much a very good replacement for traditional radar systems used in ATC. Also, ADS-B aircraft, if equipped with ADS-B IN to receive a significant amount of date that is broadcast from the ground - traffic information, weather data, NOTAMS, etc.
Normally, for sure in the USA, this data does not require a subscription fee to be paid by the user. You just have to get it installed in the aircraft. ADS-C on the other hand, can not be expected to replace ATC radars due to the long update/refresh rates. It can be used in conjunction with CPDLC in certain areas and clearances where it does not have to be closely monitored by ATC, as in over oceanic airspace and an altitude change is approved or issued and nearby traffic is not an issue. I hope that helps and did not make the confusion worse. I just finished preparing a presentation on this subject. ADS-B by itself is over 50 slides long, and that was the simple version.