What is the difference between turbojet and turbofan engines?
Turbojets and turbofans are very similar indeed: 


both are turbine engines; 
both create thrust from jet exhaust;
and both have a rotating implement in front that can be called a fan. Although in the case of the turbojet, it isn't called a fan but the compressor first stage. 




$\ $ Junkers Jumo 109-004

So what is the difference? There are five types of turbine engines, named after their main means of generating thrust:


Turbojets. The very first type of jet engine as used in the Messerschmitt 262: a compressor, combustion chamber, and a turbine. The turbine extracts enough power from the combustion chases to drive the compressor. All air flows through the engine itself, and thrust is generated by the exhaust flow only. Bypass ratio zero.
Low Bypass Turbine Engines. A bit of a lengthy, non-catchy description. Part of the airflow bypasses the main engine, and is compressed by the first stage of the compressor which has a larger diameter than the main engine tube. The turbine drives the compressor and the (low bypass) fan, exhaust flow from the turbine still generates a sizeable proportion of the thrust - inversely proportional to the bypass ratio, which is 0 &lt; BPR &lt; about 2.
Turbofans. Same as above, named after the large visible fan-like first stage. These engines are high bypass turbine, 5 &lt; BPR &lt; 15 and ever increasing.
Turboprops. Main source of thrust is a propeller, which has longer, fewer blades than a fan to limit drag penalties. The turbine drives the compressor and the propeller, some exhaust flow is still used for thrust generation - according to Torenbeek section 4.3.5, about 5 - 10% of the thrust is generated by the . BPR up to about 50.
Turboshaft. Main thrust source is a rotor or other torque requiring device, like an electrical generator in an APU. All net turbine power is applied to the drive shaft, no practical thrust from the exhaust flow.


The five types incrementally use bypass airflow to generate thrust. Increasing bypass air allows for accelerating a larger mass of air at a lower speed, generating the required thrust at a higher efficiency. But bypass air volume is inversely proportional to airspeed: the faster the aircraft, the lower the amount of bypass air that can be utilised. At supersonic speeds, bypass air is very problematic for thrust generation.

The picture below shows the Olympus engine used for propelling Concorde, a pure turbojet with 2 axes and 7 compressor stages per axis. 



The GE CF6 with BPR of about 5 depicted below was used for B747, A300 and DC10, and is a turbofan: the first stage of the compressor sticks out over the remaining stages, and bypasses most of the air inflow outside of the main turbojet engine. This bypass air is the main distinguishing feature between turbojet and turbofan. Visually, the large fan is very distinguishable of course.