As a generic rule, illegal is an act that goes against the law, one that you are liable to state punishment for.
Unlawful is an act that is incompatible with the law. For example, I always use that of an arresting policeman. He may arrest without reasonable suspicion or without following correct procedure. This renders the arrest unlawful and so it is not a legally binding act. However, it is NOT illegal i.e.: the policeman cannot be arrested for it. This is of most use with public/constitutional law. Hope this helps.