Both have the same denotation. Wordwise, desire is a bit different as it can be used as a noun while yearn needs to be changed to yearning. The difference is yearning is more of an involuntary thing. When I picture yearning in my head, I imagine a flower reaching for the sun. Desire is more of a choice thing.

you use yearn when you know that something you desire is not possible to get, kinda like wish. she yearned for his love, but he's gay. :)

If you are using them both as verbs, then they are synonyms. They mean almost the exact same thing. Yearn, I think, is a more descriptive and powerful form of the word, but that's a personal interpretation.