The real conflict of "A Clean, Well-Lighted Place" is not between two characters, but, rather, in a more abstract sense, between man and time. The story deals with characters that all have different visions of the meaning of time – the youngest man values it, but the older characters don't. The oldest character, a man near the end of his life, is simply passing the time until he dies (in fact, we learn that he even tries to commit suicide to hurry along the process). The point is, the older you get, the more time wears upon you, and the more you feel your mortality – Hemingway wants us to recognize the fact that all of us will grow old and die someday, no matter how young or confident we are now. If you were looking for an uplifting story, you may have figured out by now that this is probably not it.
The older waiter sympathizes with the old man's suicide attempt because he too realizes that life is already "nada y pues nada" (14), and death will be no different.
In Hemingway's story, as man approaches death, life becomes less and less meaningful.