Murder in the Cathedral was commissioned by George Bell, Bishop of Chichester, to critique a particular historic event: Hitler's Rohm purge, also known as the Night of the Long Knives. In this event, Hitler shocked the world when he broke the rule of law by having more than a hundred people who had become inconvenient to him murdered, including his former close friend, Ernst Rohm. Eliot's play focuses on a parallel event in British history, when Henry II in the twelfth century broke the rule of law by suggesting to his followers that he would like to be rid of the Archbishop of Canterbury, who had become inconvenient to him. The play focuses on the murder of the archbishop.
In contrast, The Waste Land describes with despair the decay of all of Western culture. If Murder in the Cathedral is meant to urge people to a principled and active response to the barbarism of Nazism or any similar barbarism, The Waste Land is a cry of mourning for the loss of Western civilization, which seemed, to Eliot, to be in shambles at the end of World War I.
Both works are alike, however, in being heavily allusive and drawing heavily on the past, using both the history and formal literary devices of prior cultures to tell a story about modern culture. Both lean into the Christian story and Christian theology.