We use both of these conjunctions when we want to communicate that one thing did not stop something else from happening. They are most commonly followed by a present participle (verb + ing).
For example;
Despite/In spite of...his bad attitude, he still got the job.
Despite/In spite of...driving as fast as he could, he arrived late.
Despite/In spite of...having bought extra food, they were hungry after the party.
Despite/In spite of...not speaking any German, they really enjoyed the museum