This is simplified, but:
ADS-B = Transponder protocol
ADS-C = SATCOM protocol and is normally used concurrently with CPDLC (Controller/Pilot Datalink Communications), but can be used without CPDLC.
Both ADS-B and ADS-C are used for Air Traffic Control, but for different purposes. ADS-C is not yet used for company communications. Most companies would like to see their flight's ADS-C reports, but that is not a functional part of ADS-C (yet).
ADS-C is typically used for oceanic or remote areas, so the aircraft's contractual reporting takes place over a longer period of time (10+ minutes typically). The time between reports is less for airspace where it's necessary (30 nm Lateral/30 nm longitudinal, or Reduced Lateral or Longitudinal Separation airspace). This allows ATC to more quickly identify the need for conflict resolution.
More than one Air Navigation Service Provider (ANSP) can see one aircraft's ADS-C reports at the same time. With CPDLC, only one ANSP can have current datalink authority (communications) with the pilots.
ADS-C is not considered surveillance control.
ADS-B is normally considered surveillance control. In fact, the aircraft's visibility to ATC is better because the aircraft's position is updated more often than most conventional radars.
You can learn more about both ADS-C and CPDLC by referring to ICAO's Global Operational DataLink Document (GOLD). Chapter 5 is a good reference for pilots, though most companies have their own slimmed-down version of Chapter 5.
http://www.icao.int/APAC/Documents/e...D_2Edition.pdf