"Although", "despite", and "nonetheless" are all words that appear in a conjunction of two seemingly conflicting expressions.
"Despite" usually starts off a noun/phrase (a group of words) : I ran outside despite the pouring rain.
"Although" heads off a clause (subject+predicate) : I let him leave, although he was my best friend.
"Nonetheless" bridges some thing that otherwise would not likely happen, and with that follows after the initially stated situation: I was confident that he would not pay me, since I did not help much. Nonetheless, he paid me three meals' worth of fare.
The three words, "despite", "although", and "nonetheless", differ in where they place in sentences ("nonetheless" is never first, and the other two are not limited this way) as well as what grammatical structure they accompany ("despite" is always with a noun/phrase, while the other two are not but rather usually come with clauses.)