Oh my, the question for Paul might be, why doesn't he kill himself?
By that I mean, Cather's story was based on a real life suicide she knew of, and so in some ways her portrait of this young man was always aimed at his death. However, it's also possible to look at how out of place he was in his world his whole life, and say, again, he was always aimed at death. He was suspended from his high school, and so was in some ways shamed/outcast by what he'd done. The second paragraph of the story mentions his drug addiction; drug addiction leads to poor judgment. However, though all of this is true and applies, the main reason he killed himself was that he hoped for a reason to live through his contact with beauty/art. When that was disrupted, he was lost.
After his crime got exposed to the people around him, he has no guts to face them and cannot bear to return back to the "ugliness and commonness" of Cordelia Street. Thus. he committed suicide by jumping in front of a moving train, killing himself instantly, giving the ending a tragic note.