In my experience, these words are the same. But in the US we don't usually use "despite." It sounds formal and old-fashioned. But you will still see it written in many books.
Despite her bad manners, I gave her good directions.
I was glad the plane had a safe landing in spite of the rain.
She wore a bright red dress to the party, in spite of the invitation saying "wear black and white."
Now you try some examples and I want to see how they sound. Despite your lack of confidence, you should still try!