In the 1930s, doorbells that rang like an alarm were considered at best, annoying, and at worst, as dangerous to ones health. After electric lights, doorbells were often the second residential electrical appliance in a home. Early doorbells were exactly that: bells which sounded like a fire alarm. Thus door chimes were marketed to combat "Public Noise Enemy #1" and calm "Doorbell Nerves".
Sure, early doorbells alerted you that somebody was at the door but it was like hitting you over the head with a hammer!. In the 1930s door chimes saved the day by treating the world to a much more pleasant sound by striking one or more tubular bells or metal xylophone-like tone bars.
Today, door chimes produce the ubiquitous "ding-dong" sound we know today. For more on the subject of door chimes vs doorbells, please see the article on my sister site, the Doorbell Museum.