I don’t think there’s any perceptible difference in the meaning and usage of “in spite of” and “despite.” They are both prepositions that mean “in defiance or contempt of” or “without being prevented by.” Consider the following two sentences:
“We went to the park in spite of the rain.”
“We went to the park despite the rain.”
These two sentences mean exactly the same thing, without any perceptible difference of nuance. “In spite of” and “despite” also have exactly the same language register, which means that that either can be used in any social context—whether the language is informal, formal, legal, scientific, or scholarly—without any difference in denotation.
The only notable difference between them is, of course, structural: “in spite of” always comes with the preposition “of,” while “despite” doesn’t need “of” to work. That’s all.