The obvious difference between a personal essay and narrative writing is that the first is autobiographical -- and must therefore be true -- while the other is fiction. The confusion comes with the element of verisimilitude, the "quality of appearing true or real" in fictional writing. True personal writing tells the truth; narrative writing may contain many true-to-life elements, but it will still be fiction.
Personal Essays are Relaxed, Like Some Narratives
Personal essays have much in common with narratives. Personal essays are "wild children" of composition; loose structure and prose experimentation are practically de rigeur, just like some narratives, while style and point of view can be exactly the same as fictional pieces. Grammar and subject matter, according to Annie Dillard, are "all over the map." These essays cannot be done incorrectly except they be done falsely, as fiction writers who faked autobiographies -- like James Frey -- discovered. Autobiography and personal essays, to be such, must be true.
Truth is Premium
If truth is the only requisite for a personal essay, it's not odd that these pieces go cheek-by-jowl with the genre piece. Narrative pieces need not follow rules, either -- as writers from Shakespeare to James Joyce have demonstrated -- except that they tell their stories well, are entertaining and are truth at least to a degree. There's the rub; even Stephen King in "On Writing" tells authors to write about "anything at all ... as long as you tell the truth" -- meaning, of course, truth as the writer sees it.
Narrative Truth is "True to Life"
This is a real difficulty for some, since narratives from many authors read so "true to life" that the reader's desire to make them autobiographical is irresistible. Kipling spent seven years in India before returning to England to write, yet many believe he was a life-long resident because of his works; Edgar Allan Poe is frequently considered insane, simply because of his "mad" narrators. And the aforementioned King confesses that his works give a terrifying patina to the life of the mildest of men.
Two Kinds of Truth
So, despite the confusion, the only real difference between the genres is the use of literal truth in the personal essay, which must be scrupulously real to maintain its integrity, and the quality of verisimilitude, the appearance of truth that can give reality to the purest narrative fabrication. Both pieces should be true to the author's ideas, but only the essay must remain true in every detail.