What is the difference between the titles Frankenstein and Frankenstein; or, The Modern Prometheus? Which is the correct title?
Mary Shelley's novel was written under the title: Frankenstein; or The Modern Prometheus. As used to be the case with many pieces of old literature, the title may have been long, eventually to be shortened later. In a modern context, I see this novel referred to most often as Frankenstein, but it was published under the longer title in the early 19th Century. Shortening titles is not at all unusual.
For example, William Wordsworth's poem "Lines written a few miles above Tintern Abbey" is often abbreviated to read simply: "Tintern Abbey." While we refer to Shakespeare's famous play as Hamlet, its original title was The Tragedy of Hamlet, Prince of Denmark. Mary Shelley's novel is called Frankenstein to make it easier to refer to, and as may also be the case with the advent of typesetting, typing, etc., easier to put into print (and less expensive with less letters and ink)—when referred to by its shortened name.
Started when she was 18, Mary wrote her tale in response to a challenge by Lord Byron to write a scary story.
Mary seems to be the only person present that night that published a story. Some sources note that she was inspired by a haunting dream. Some say she dreamt of a monster created by man; others report that in her dream, her baby, who had died in childbirth, came back to life—one way or another, it was in this context that Frankenstein was "born."
The allusion to the mythical character of Prometheus is found in his creation of mankind:
According to [some] sources, Prometheus fashioned humans out of clay.
Because Prometheus created life, Shelley saw a parallel between Victor Frankenstein's experiment, in which he harnessed energy that enabled him to bring the creature he had built to life, and Prometheus' creation of humans from clay—into which he had breathed life.
So Shelley's title accounts for the parallel she saw between the mythological Prometheus and her character of Victor Frankenstein; so she gave it a title that foreshadowed a fundamental aspect of her classic novel.