'Nevertheless' is an adverb, and it stands alone in a sentence. It can go at the beginning, middle or end of a sentence, usually separated from the rest of the sentence by a comma (,). Its meaning and use is similar to the word 'However'.
'In spite of' can't stand alone. It has to be followed by a noun, or a verb in the form of a gerund:
In spite of my illness, ....
In spite of being ill, ...
It has the same meaning and use as the word 'Despite'.