What is the difference between a propeller and a rotor?
Short answer:


A propeller is used to propel something, the force at stake being called thrust. Thrust is usually horizontal (but not always, think about rockets). This device comes from the navy domain.
A rotor of a rotorcraft is used to control anything, including the altitude, the generated force being called lift. Lift is generally vertical (but not always, think about a rudder). Rotors have generally more than two blades.


Lift and thrust involved are components of the same physical force (the aerodynamic force) to which we refer to differently based on their purpose after an arbitrary breakdown based on the direction they act (compared to the direction of the aircraft or to the gravity field).

For our pleasure, let's also talk about a turbine rotor, which is a rotor/propeller which doesn't produce lift or thrust, but convert the fluid velocity (actually momentum) into torque. For example the wind turbine rotor. A turbine rotor is certainly a propeller, but we don't refer to it as a propeller, as it propels nothing.



Rotor is the most generic name for something spun by a shaft. It can be used for a propeller, the rotating wing of an helicopter, the rotating elements of a compressor or a turbine, etc. You can't be wrong using rotor for any spinning device. Rotor (rotating) is the antonym of stator (stationary). These terms are not specific to aerodynamics, in a electric motor there is usually a stator and a rotor as well.

The rotor of a rotorcraft comprises blades and is comparable to a propeller, if we except blades additional freedom in flapping and leading-lagging to limit vibrations created by repeated force swings encountered while performing a 360° turn.


Source: Wikipedia

But actually the main rotor is a (rotating) wing, and the anti-torque rotor is a (rotating) rudder. Their blades are meant to produce lift like the wing and the rudder they replace. Because lift is involved, we tend to name them rotors rather than propellers.

A propeller also comprises blades, fixed on the rotation axis.


Source.

The purpose of a propeller is to propel the aircraft, that is to push it in the direction of its flight.

Both a rotorcraft rotor, a propeller and a wing create the same aerodynamic force. However the conventions are:


A rotor and a wing are used to eventually counter weight and control altitude, and a propeller to eventually counter drag and control velocity along the flight path (I say eventually because to gain altitude you usually temporary increase thrust, that's a detail)
A rotor and a wing create lift, a propeller creates thrust.


Wing, propeller blade and rotorcraft blades work exactly the same from an aerodynamic standpoint. The resultant of their action is the total aerodynamic force, which is then broken down arbitrarily into lift, drag, thrust and torque to facilitate mechanical analysis.

Notes:


Moving an aircraft along its path (thrust) is easier than maintaining its altitude (lift), and therefore a main rotor is far larger than a propeller for aircraft of the same mass.
The horizontal displacement of the rotorcraft is also produced by the main rotor (cyclic control), so some part of the aerodynamic force can also be seen as thrust, and the rotor also acts as a propeller.
Propeller is used for simple propellers and not in turbine engines. While the fan of a turbofan is comparable to a propeller, participates to thrust production and aircraft propulsion, the fan is still only one part of a system (guide vanes, fan duct and exhaust nozzle greatly maximize final thrust). Bottom line the fan is a rotor, not a propeller.