The FAA utilizes the ICAO definitions for class A-E airspace as controlled and class G as uncontrolled (class F airspace is unused under the FAA). What are the key differences between controlled and uncontrolled airspace in general?
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.
The shortest answer is: in uncontrolled airspace, you fly IFR without a clearance (and no clearance is available); in controlled airspace, you cannot fly IFR without a clearance.
That is the only real difference. Everything else (like weather minimums for VFR) follows from that.
BTW Class E is controlled airspace, and has no specific entry requirements for VFR either.