What is the difference between believers (mu'minin) and Muslims (muslimin), and which has a higher rank?

The meanings of "Islam" (submission) and "Iman" (faith) differ depending on their context. If they appear separately, each denotes the entirety of the religion. However, if they appear together, "Islam" refers to outward actions, whether accompanied by inner faith or not, whereas "Iman" refers to inner, heartfelt actions that only a true believer performs, making Iman a higher rank. An example of this is the story of the household of Lot, where the believers (the survivors) were brought out, and Allah found "none but one house of those who submitted" (which included Lot's wife, who was outwardly Muslim but inwardly a disbeliever), thus affirming the distinction between the two concepts.