An ‘unhappy individual’ may choose not to talk about the cause of his unhappiness. He may keep his thoughts to himself, and may do nothing to change the situation that he is in.
A ‘malcontent’, on the other hand, will not keep quiet. He not only complains about the situation he finds himself in, he also tells you what he is going to do to solve the problem. A malcontent is a man of action; he is a rebel who is willing to fight those in power.
While an unhappy individual focuses on personal problems, a malcontent focuses on the problems he sees in the society we live in, it could be social or political injustice. Unlike an unhappy individual, he wishes to bring about change; not necessarily, by peaceful means. He is someone who is very difficult to deal with.