What's the difference between the professional titles "writer" and "screenwriter"?

From What's the Difference?: How to Tell Things Apart that Are Confusingly Close

• 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.

• An ampersand (&) indicates multiple people or teams who wrote together.

Example:

Pirates of the Caribbean: The Curse of the Black Perl

• Screen Story by: Ted Elliott & Terry Rossio and Stuart Beattie and Jay Wolpert

• Screenplay by: Ted Elliott & Terry Rossio

In this case Jay Wolpert was hired by Disney to write a script based on the theme park ride. Later Stuart Beattie was brought in to rewrite the script. After that Ted Elliott & Terry Rossio worked as a team on another rewrite.

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The Writers Guild of America (WGA) has a page with more details on Screen Credits.