[subject] + 因 + [noun / short phrase] + 而 + [verb phrase]
[subject] + 因为 + [clause] + 而 + [verb phrase]
因…而 puts a cause and its result inside a single clause. 因 introduces the cause, and 而 links it to the verb that follows. It's written Chinese, common in news, notices and essays, and the cause is often only one or two characters long.
With a short cause, 而 can drop out without changing the meaning: 因雨推迟. Keeping it makes the line sound more formal.
In less formal writing 因 becomes 因为, and the cause can be a whole clause.
A few fixed phrases use the frame: 因人而异 "varies from person to person", 因地而异 "varies from place to place", 因时而异 "varies with the times".
English puts "because of the rain" at the end. Here the cause sits between the subject and the verb.
因而, written as one word, is a different item: a conjunction meaning "consequently" that opens the result clause.