Why is one side of a dune different from another?

Because the wind blows the sand one way, not both.

One side of of a sand dune is different than the other side because each side is formed by a different element. If one side is formed by wind and the other side is formed by water, it is impossible for the two different sides to be identical.

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First sentence correct; second, not so. Although water does deposit sand in dune-like ways, the question is about sand-dunes, formed only by the wind (which is not an "element" either; nor is water). Wind and water will not work as suggested. The simple answer is that as the sand ridge builds up the wind blowing over it eddies over the edge, dropping sand grains close to the lee slope.