Interesting question... they tend to be used interchangeably, somewhat in error, but still communicates meaningfully.
In common usage, composition is the proportion of different parts to make a whole while constitution is the act, or process of setting something up, or establishing something; the composition or structure of such a thing; its makeup.
composition 1) something's ingredients like a budget/cake 2) creative work like an opera/poem
constitution 1) composition/ingredients like atoms in a chemical 2) document establishing an organization like US 5th amendment
Perhaps I should "plead the 5th" answering this question :)