(Note: While there are locations mentioned that are very New York, to answer this question, I believe it's important to look at what happens to the characters while in the city.)
New York is seen as a very different place in The Great Gatsby and in The Catcher in the Rye. In Gatsby, New York is a place of dreams; it's a place where "Anything can happen" ... "Even Gatsby could happen." However, in Catcher, New York is the place Holden ultimately suffers a mental breakdown. In both novels, New York is a destination though. Holden wants to go to New York to hang out after being expelled from school and it's where Nick goes to work as a bondsman.
In both Catcher and in Gatsby, New York is the place where Gatsby's false dream of Daisy and Holden's struggles reach their climax. In New York, on the hottest day of the year, Gatsby's dream of having Daisy dies. After arguing with Tom in New York's famous Plaza Hotel, Gatsby (subconsciously) realizes that he will not end up with Daisy as she is appalled that he is a bootlegger. Nick describes Gatsby this way:
"[W]ith every word [Daisy] was drawing further and further into herself, so [Gatsby] gave that up, and only the dead dream fought on as the afternoon slipped away, trying to touch what was no longer tangible, struggling unhappily, undespairingly, toward that lost voice across the room."
In Catcher, Holden's ultimate psychological breakdown occurs while walking through Fifth Avenue. He describes this event as "something very spooky" and then goes on to describe his break this way:
"Every time I came to the end of a block and stepped off the goddam curb, I had this feeling I'd never get to the other side of the street. I thought i'd just go down, down, down, and nobody'd ever see me again."
The Catcher in the Rye goes into much more detail about specific New York places. Holden describes the Museum of Natural History in detail and he also spends a lot of time in Central Park, particularly the carousel and the frozen pond where the ducks no longer reside. Both of these descriptions, however, have to do with Holden's desire to keep things the same or to help maintain the innocence of children, particularly his sister, Phoebe.