It is also significant that the flowers represent artificiality:  the flowers Paul sees as he rides through New York are growing under glass, in an artificial environment.  Paul tends to find natural things (or flowers in their natural state) ugly and observes that "the natural nearly always wore the guise of ugliness." This is significant because Paul can't love himself as he is.  Rather, he loves and nurtures the artificial illusion of himself that he creates.

The flower in Paul's lapel is his attempt to be something he is not--to appear flippant and "better than" his teachers, to show that he doesn't care about being disciplined.  Later, when the flower begins to droop, it is as if the brief period of glory Paul experiences is also ending.  He remembers the flowers under glass and thinks that they, too, have faded by now, having experienced evanescent glory. His final act of burying the flower under the snow, of course, foreshadows Paul's own death.

Paul believes that he appears more beautiful in his new attire and way of living though it is all false--artificial.  The tragedy of this story is that Paul is never able to learn who he really is and to love that person.

The flowers that appear consistently in the story represent Paul trying to see and find beauty in a very mundane, uncertain world.  When he wears the red rose when he appears before the faculty, it stands out in a negative way, appearing garish and tacky instead of beautiful and subtle.  Another example is when the carnation wilts towards the end of the story.  He buries the carnation in the snow, which can be symbolic of his own death.  He is burying himself in many ways, along with this hopes and his dreams:

They also symbolize Paul, who, like flowers in winter, is out of place. The flower-killing snow Paul sees on the train to New York and by the railroad tracks at the story's end provide a stark contrast to the bright flowers Paul surrounds himself with. (eNotes)