Harmony refers to a way of combining similar elements (line, shapes, color etc.) in an artwork to accent their similarities. It could be accomplished by using repetition and gradual changes. Pieces that are in harmony have an overall uniform appearance.
Variety is a way of combining elements in ways to create complicated relationships. A picture may have a variety of hues, values, lines, textures and shapes.
A picture made up of many different hues, values, lines, textures, and shapes would be described as complex. Complex pictures increase visual interests.
However, keep in mind that harmony and variety are not opposites. A careful blend of the two principals is essential to the success of almost any work of art.
Artist who only focus on harmony and ignores variety might find it easier to achieve balance and unity. However, visual interest in the piece would be lost.
On the other hand, artists who only focus on variety and not harmony would make their works too complex. As a result, the viewer would be lost and confused. Furthermore the overall unity of the piece would be lost.