This a great question we had a while back, the obvious difference between an amateur and a professional artist is the professional artist makes a living from either selling or demonstrating his artwork.
However, there are very good amateur artists who do sell their work and they can be classed as professional, although I personally would prefer the term semi-professional.
But in truth there is always a dividing line between the two and this really can only be seen by someone with a good eye for art. The average person in the street would find it difficult to discern between a good amateur painting and one done by a professional.
This is the reason it rankles the professional when an amateur painting is sold for a lower price when it is obvious to the professional that the standard of the painting is lacking that special touch, but the client does not see this difference.