"Herald Square” was named after the New York Herald newspaper in 1893. It is the area around West 34th Street.
“Times Square” was named after the New York Times newspaper in 1904. It is the area around West 42nd Street in Manhattan. A popular math joke is: “Where do math teachers go on vacation?"/"To Times Square.”
Other squares named after newspapers include Globe Square (after The New York Globe), Telegram Square (after the New York Evening Telegram) and Times Plaza (after the Brooklyn Daily Times).
Wikipedia: Herald Square
Herald Square is formed by the intersection of Broadway, Sixth Avenue (officially named Avenue of the Americas) and 34th Street in the borough of Manhattan in New York City. Named for the New York Herald, a now-defunct newspaper formerly headquartered there, it also gives its name to the surrounding area. The intersection is a typical Manhattan bow-tie square that consists of two named sections: Herald Square to the north (uptown) and Greeley Square to the south (downtown).
Wikipedia: Times Square
Times Square is a major commercial intersection and neighborhood in Midtown Manhattan, New York City, at the junction of Broadway and Seventh Avenue, and stretching from West 42nd to West 47th Streets. Brightly adorned with billboards and advertisements, Times Square is sometimes referred to as The Crossroads of the World, The Center of the Universe, the heart of The Great White Way, and the “heart of the world”. One of the world’s busiest pedestrian intersections, it is also the hub of the Broadway Theater District and a major center of the world’s entertainment industry. Times Square is one of the world’s most visited tourist attractions, drawing an estimated fifty million visitors annually. Approximately 330,000 people pass through Times Square daily, many of them tourists; while over 460,000 pedestrians walk through Times Square on its busiest days.
Formerly Longacre Square, Times Square was renamed in 1904 after The New York Times moved its headquarters to the newly erected Times Building, the site of the annual ball drop which began on December 31, 1907, and continues today, attracting over a million visitors to Times Square every New Year’s Eve.
19 August 1893, Trenton (NJ) Times, pg. 5:
The New York Herald Moves.
NEW YORK, Aug. 19. The Herald today contains the following announcement. “On Sunday, Aug. 20, the New York Herald will remove from its present quarters at Broadway and Ann street to its new building in Herald square.
28 January 1894, New York (NY) Times, pg. 10:
Broadway and 35th St., Herald Square
23 March 1904, New York (NY) Times, pg. 8:
To the Editor of The New York Times:
When the new building of THE NEW YORK TIMES shall be completed and become a thing of art and beauty in that section of the city in which it is to stand, why would it not be fitting that the space about the edifice be called “Times Triangle,” or “Times Square,” although perhaps it may not be a square?
It is, it seems, more euphonious than “Long Acre Square,” and very soon would become as well known as “Printing House Square” or “Herald Square.”
No doubt the Board of Aldermen would take up such a suggestion at the proper time and act upon it favorably. Can it not be entertained?
J. W. C. CORBUSIER.
New York, March 17, 1904.
By Samuel Marx
New York, NY: Donald Flamm
You may have never heard the name of some of them. But they are part, particle and parcel of New York’s Main Stem, Hardened Artery, Great White Way, Incandescent Lane, Mazda Boulevard, Chow Mein Stem, Double Cross’Roads of the World, Two’Times Square, or, as somebody once called it, Broadway.