There was a mention of this today on the Leonard Lopate Show on WNYC (can be gotten on line @ WNYC.org). It agrees with the New Orleans Stablehand interpretation and adds that they called New York the Big Apple because the tracks in New York had the largest purses. The first appearance in print was as the title of a Newspaper column in the early 20's.

The nickname "The Great White Way" for Broadway came from the amount of lighting put there by the theater district. It is my understanding that Broadway was the first New York Avenue to be lit by electricity for its entire length.