Most dictionaries suggest that inspite and despite are synonymous, but are there any specific instances when their usage is not interchangeable?
I would suggest that inspite — as written in your question — is not in fact a word.
I think you must mean in spite of, which is directly interchangeable with despite.
He went for a walk in spite of the rain.
He went for a walk despite the rain.
I am not aware of any real difference between the two options, though I tend to use despite purely for efficiency of words.
It is true that "inspite" is not a word. The difference between "in spite of" and "despite" is more in connotation than in efficiency:
"In spite of" usually connotes a degree of contempt or rebellion. For example, one could say, "In spite of the supervisor's mandate, Pauline went out for lunch." This suggests that Pauline does not think highly of the supervisor's mandate.
"Despite" is usually more of a neutral contradiction. "Despite the supervisor's mandate, Pauline went out for lunch." This shows that the supervisor has not affected Pauline's plan for lunch either way; she just doesn't care.
These ought not be confused with the infinitive verb "to spite", which is meant as a direct rebellion: "I punched him in the face to spite him."