An exposé is a piece of journalism that exposes some situation or occurrence. A damning expose is one where specific companies or people involved in the situation are shown to have behaved in a very bad way - often illegally. An example might be where someone exposes evidence of organised paedophilia in a church, and where the higher officials have knowingly covered it up.

usually used to mean a piece of writing that reveals the guilt of someone or something. For example, "That article on Rupert Murdoch was a damning exposé of his complicity in the News Corp. phone hacking scandals."

An exposé is an investigative news article/interview/documentary trying to literally EXPOSE secretive details of a person's life (or, less frequently, details of a private event/ideological movement). Something is damning when it does extreme, irrevocable damage in such a way that mimics the torture of the Christian/Muslim hell. So a damning exposé is something that a journalist did to uncover details about someone or something, and those details are so grotesque or shocking that the person/thing can never recover from them. Example: "After the damning exposé on Senator Smith's private life revealed that he'd been frequently soliciting transgendered teenage prostitutes in Thailand for over a decade, he resigned in disgrace and fled the country."