Although is a conjunction, which means that it introduces a clause (ie, it ultimately targets a verb): Although I knew the answer, I didn't raise my hand.
Despite (also: in spite of) is a preposition and prepositions come before (or ultimately target) nouns (or pronouns, or gerunds - which are actually verbal nouns): I chose to go out despite the bad weather. - Despite his good intentions he failed to convince them. - Despite feeling (= although he felt) rather depressed, he decided to go to his ex-girlfriend's wedding.
Tip: despite the fact that = although: Despite the fact that he was ill, he went to work = Although he was ill, he went to work.
Although could go with a gerund if it were part of a complete clause:
Although going to the cinema seemed a nice idea, we decided to stay at home. Although goes with clauses or adjectives, but doesn't fit into thie example in the poll.
what you mean by "part of a complete clause"? why in "although feeling sick", although is not part of the clause?