Here is one of my favorite entries from my book What's the Difference?: How to Tell Things Apart that Are Confusingly Close, modified only slightly from its publication version:
What’s the difference between “STORY BY,” “SCREENPLAY BY,” and “WRITTEN BY” in film credits?
Hollywood filmmaking is a collaborative art that begins even before the first scene is shot. Even before the actors are hired. Yes, even before merchandising rights are sold. A screenwriter will tell you that, creditwise, he’s probably going to be screwed in one of two ways. One, either his name will stay on a film he originated or wrote an early draft for although subsequent writers rewrote it so much (and often, so badly) that none of his work remains. Or two, he’ll be the rewrite guy who molds the screenplay into brilliance but who won’t get credit due to one of the following policies, each of which carries its own formulas of just how much of a script a person needs to contribute (and when) to earn that particular credit.
A “story by” credit is given to the person or team who came up with the essence of a film (such as the plot or main characters) and who may have written a treatment, but who didn’t write the screenplay. Similarly, a “screen story by” credit goes to a person or team who adapted other material such as a novel, a TV show, or a news article for film and made it substantially different from the source.
A “screenplay by” credit is given to the person or team who wrote the scenes and dialogue of a screenplay but didn’t generate the idea for the story.
A “written by” credit is given to the person or team who both conceived of the story and wrote the screenplay. It usually merges “story by” and “screenplay by.”
“And” indicates multiple writers or writing teams who contributed but did not collaborate directly—they may never have even met. Examples are “Andrew Douglas and Justin Goldstein” or “Mike Fox and Bethany Kant and Rachel Loonin.”
An ampersand (&) indicates multiple people or teams who wrote together. Examples are “Darren Sapper & Matt Small” or “Kevin Alansky & Rachel Fremont and Seth Kessler & Dara Neumann.” (Here, Alansky and Fremont wrote together, Kessler and Neumann wrote together, but the two pairs did not make it a foursome.) You might also see a team and an individual such as “Mike Chasen & Randi Skylar and Chris Campagnuolo.”