What are the key differences between controlled and uncontrolled airspace under the FAA jurisdiction?
This Document covers all the regs pretty well. But in short the key point on class G (uncontrolled) is 


  Although ATC has no authority
  or responsibility to control air traffic, pilots should remember
  there are visual flight rules (VFR) minimums which apply
  to Class G airspace.


Class G airspace also holds no specific entry requirements (aside from VFR minimums), no specific aircraft equipment (no radio, no transponder etc) and no minimum pilot certificate requirement (but you still need a license (one of a few varieties) you just don't need anything in addition like the instrument rating requirements for class A). On the contrary glass G airspace (uncontrolled) is not a free for all and you are not free to do as you please in some regards. While ATC has no reach into Class G the FAA does and you are still bound by the regulations the FAA puts forth that apply to class G airspace.  

Likewise the definition they provide for controlled is  


  Controlled airspace is a generic term that covers the
  different classifications of airspace and defined dimensions
  within which air traffic control (ATC) service is provided
  in accordance with the airspace classification.


If you are wondering what the specific purpose of ATC is they define that as well, 


  The primary purpose of the ATC system is to prevent a
  collision between aircraft operating in the system and to
  organize and expedite the flow of traffic.