I don't think there is any difference in meaning. The two can be used interchangeably, though Alex is right that "despite" sounds just slightly more formal. After "despite" or "in spite of," you can use either a noun (or noun phrase or pronoun) or a gerund (or gerund phrase). For example: "Despite her fear of heights, she climbed up the ladder." ("her fear of heights" is a noun phrase; the main noun is "fear") "Despite being afraid of heights, she climbed the ladder." ("being afraid" is a gerund phrase) "In spite of having seen the movie twice, he did not understand the story." ("having seen" is a gerund phrase) "In spite of the cold, they went outside." ("the cold" is a noun phrase; "cold" is the noun) "Despite the fact that he knew the answer, he did not raise his hand." ("the fact that he knew the answer" is a noun phrase; "fact" is the main noun) (Edited: I originally said "participle" instead of "gerund," but I've checked some grammar sources and it seems that the -ing phrases after "despite" are gerunds, not participles. In any case, you need a verb that ends with -ing. The -ing word will usually describe the subject of your sentence, which makes it seem more like a participle to me, but you can't use participles that end in -ed. You need an -ing word. So it is probably correct to regard these as gerunds.)

Thought provoking answers. I disagree that these are too formal for speech. Also, there is an idiomatic expression “doing something in spite of himself” where I don’t think you can use “despite”. “I usually hate going drinking but I had fun last night in spite of myself” “Despite being tired, I kept driving.” Personally I would not use “in spite of” here. In general I would favor “despite” before a gerund. “In spite of her good looks, she didn’t have much experience in relationships.” “Despite her good looks, she didn’t have much experience in relationships.” (About the same)

They mean the same thing. "Despite" is just a bit more brief. You typically see these phrase used more in writing than speaking. Example: Despite what he said, I'm still going to buy it. In spite of the fact it cost more, I'm still buying it. Despite what my parents said, I'm still.... In Spite of the fact my parents said, I'm still.... The word "spite" means to go against.

They are different. 'In spite of' usually means going against someone/something, like you don't care about it. 'Despite..' is more normal, just like 'even though...'

There's no difference, you can use either. The only rule is that after those two you cannot use a subject and a verb, which means you can't say: "Despite I was busy I still angeed to help them." You have to say: "Despite being busy...". So, you can use a noun (without a verb after), a participle.

Have a look Murphy book, there is a unit about it.