Natural harmonics are produced when you play harmonics on an open string: these harmonics generate the overtone series corresponding to the frequency of the open string.
With artificial harmonics, the harmonics are played on fretted notes. E.g., if you fret the 6th fret on the G string and place your thumb on the 13th fret and pluck behind it (i.e. towards the bridge), you get an artificial harmonic which is a fifth above the fretted note. The fretted note was C# and the harmonic you obtain is G#.
So, with natural harmonics, you're limited to the harmonics that feature in the overtone series of the open strings' frequencies (i.e. only the harmonics produced off the notes E, A, D, G). OTOH, with artificial harmonics, you can generate harmonics based on the overtones relating to any note you fret and not just E, A, D, G.
Hope that helps
