The story "Paul's Case" was based on the real suicide of a high school boy (as discussed on my Willa Cather biography page) that Cather heard about when she was a teacher at a Pittsburgh high school. It is probable then, that this is the reason she set the story in Pittsburgh.
Also, the grimy streets of Pittsburgh, covered in soot, dirt, and ash from the many steel plants in town, provided a stark contrast of the clean, well-lit, beautiful streets of New York City. Cather describes a setting from which ANY boy would want to escape.
Cordelia Street, and Paul's house in particular, have an aura of despair, hopelessness, and futility about them. It isn't so much that the description of the street and house have a desolate mood surrounding them. It would be better to say that they are described so as to cause one to FEEL despondant, depressed, or helpless to change the situation. It almost makes the reader want to reach into the story and pull Paul out of Pittsburgh and drop him into a better life.
Then Paul finally succeeds in his quest to live a better life. He travels to New York City, on stolen money, and lived up the good life. Cather described New York City as a haven. Of course, Paul was on Fifth Avenue, so he didn't see any of the desolation that exists in New York City, which provided even more of a glaring contrast to Cordelia Street. Fifth Avenue had a feeling of freedom, and adventure in every lamp post, street, taxi, shop, and hotel.
I think that Cather chose Pittsburgh because it has a connotation of
being dirty, scummy, and vile. On that other hand, New York City is the
city that never sleeps. At the time of the story, it was seen as glitzy,
galmorous, and clean.
The bulk of "Paul's Case" was set in winter. The most significant aspect of winter in relation to Paul is that flowers don't bloom in winter. It sounds strange to pick that out of all the things that winter could mean, but flowers represent a lot to Paul. In the beginning of the story, he wears a red carnation in his buttonhole to show his defiance of the panel of faculty members condemning him. He thinks his house unworthy of flowers, but his hotel room is. He passes a flower shop while out one night, and (here's where that significance part comes in) thinks about the flowers blooming despite the winter, in defiance of it even.
Paul's Life can be considered a winter. Very few "flowers" have grown in his wintery life. By flowers, I mean those little things, and the big things, that make life worth living, such as love, a smile from a stranger, and unprovoked bouts of joy. Paul doesn't have many of these things while in Pittsburgh so he doesn't have flowers except to defy the wintery stares of his teachers.
When Paul travels to New York City, his life begins to bloom. He decorates his room with flowers, he wears them in his buttonholes constantly, and he begins to see the flowers around him. He wines, dines and parties while in New York City and gets lost in the music of a hotel lounge quartet, just as he did while listening to the symphonies at Carnegie Hall. However, this time Paul enjoys it even more because he knows that when the song is finished, he doesn't have to go back to the despair on Cordlia Street. New York City, even in the throes of winter weather, becomes Paul's summer.
Paul's summer is a sharp contrast to the cold crispness of the New
York winter. Cather shows the reader how warm joy can make you feel,
regardless of the coldness of the weather.