Betrothal carries two implications for me - first, that it wasn't the decision of the people involved, and was perhaps a decision taken when one or the other was still a child. And second, that while there is a promise of marriage, there's no date or certainty to it.
In contrast engagement means "we're going to get married". (There was once the idea of "engaged to be engaged" - people who were still in school or otherwise too young to get married or even plan a wedding, but wanted to say they would get married some day.) Old fashioned people like me say you're not engaged unless you have "a ring and a date" but I know plenty of people who use the words fiance and fiancee to mean "person I love who I will probably end up marrying" and in many cases "person I love, live with, and am raising children with who I will probably end up marrying" but they have no firm plans to actually do that. Language moves. You won't go wrong by assuming a fiance or fiancee means there is an actual wedding in the next year or less, but don't be shocked if it doesn't and the person just wanted a stronger word than boyfriend or girfriend.
I usually only hear the -to-be forms during actual wedding planning conversations with people who don't know both parties. "Where will you and your husband-to-be live after you're married?" for example. This solves some time-travel problems that would otherwise be in the question: "you and your fiance" won't be living anywhere after you're married, because he won't be your fiance then, but he's not your husband now which makes the question odd if someone just asks about "you and your husband". The -to-be fixes that. Like when people say "I was on vacation with my then-fiance" if by the time they tell the story, he's been promoted to husband.
Of course if they know you, they can just ask "Where will you and John live after you're married?" so it has a more formal use. Perhaps because of that, the -to-be forms are more formal to my ears than fiance and fiancee. They sometimes appear in advertising: "everything for the bride-to-be" for example as well as in conversations. A bit like mother-to-be, they're not everyday speech unless someone is being amusing by mocking that kind of talking.