there is no difference between them. they show contrast and surprising situation. Only, "In spite" must be followed by "of" , while "despite" is not followed by "of". the key point is "of". Moreover, they must be followed by noun or verb in gerund.
for example: In spite of the bad weather, the boy went out.
despsite the bad weather, the boy went out.
I am not an english teacher but I give my explanation of what I know. As for your sentence, I dont understand what you mean by "under this condition" ?