Deutsche Frage

The main difference is that 'Backe' starts with a 'B' and 'Wange' with a 'W'....... haha, just kidding..
In fact, there is no real difference, both words prescribe the same part of the body (the cheek).
My guess would be that it depends on the region in Germany, if one word or the other is more commonly used.

As a side note - however, when you talk about buttocks, you would use (Arsch-)Backe, but never 'Wange'....

If you want to be sure, you should use "Wange" for cheek, unless it is clear from the context that you couldn't possibly be talking about buttocks... As poster above said, "Backe" can refer to both.
"Kann ich dir einen Kuss auf die Backe geben?" (Can I give you a kiss on the cheek / buttock) could lead to embarrassing situations.

"Wange" is a bit more formal/poetical, "Backe" is more colloquial. When speaking of cheeks, both mean the same.
For children's cheeks, one can use "Bäckchen", but not "Wängchen" (that word doesn't exist).

For the rest, see kilo_fox's post.