Difference between harmonic motion and simple harmonic motion? The name simple harmonic motion suggests that its the simple version of "harmonic motion".
Does harmonic motion exist and if so is there a difference between these 2 terms?
 A: Harmonic motion would just be any motion that is periodic. For example, motion following a kind of square wave would be harmonic.
Simple harmonic motion is motion that is specifically sinusoidal; that is, it can be described by a sine wave. The reason simple harmonic motion is "simple" is because any kind of harmonic motion can be constructed as superpositions of SHMs of different frequencies, via the Fourier transform.
A: The OP should perhaps have asked "Does simple harmonic motion exist?" given that it is harder to find examples of simple harmonic motion than harmonic motion.
Any motion which repeats itself can be called harmonic motion and so simple harmonic motion is in the set of harmonic motions.  
One way of defining simple harmonic motion is via the differential equation 
$\dfrac{d^2X}{dt^2} = - \omega^2\,X$ where $X$ is some parameter which could be a displacement, an angle etc.  
In mechanics it can be interpreted as the motion of a mass acted on by a restoring force which is proportional to the displacement of the mass from a fixed point.  
Simple harmonic motion has important properties, for example, the period of oscillation does not depend on the amplitude of the motion and lots of systems do undergo simple harmonic motion even if sometimes it is an approximation.
A good example of the difference between harmonic motion and simple harmonic motion is the simple pendulum.
The simple pendulum undergoes harmonic motion, however for small angles of oscillation it does approximately undergo simple harmonic motion.
