Using Hemingway's own words, the young waiter is "the waiter who was in a hurry," while the old waiter is the "unhurried waiter." That is, the young waiter exists in the present tense of his youth; he does not sympathize with the old man as does the older waiter.  The waiter in a hurry wants the old man to leave so that he can go home to his wife; he has a full life that leaves him no time to contemplate existential questions as do the old man and the older waiter.  The older waiter tells the other waiter,

"You have youth, confidence, and a job," the older waiter said."You have everything....."

"I am of those who like to stay late at the cafe," the older waiter said..... "With all those who do not want to go to bed. With all those who need a light for the night."

Like the old man, the older waiter understands the existential angst that comes with age. Disillusioned in life, the older men understand the "nada," the nothingness of most existence:  "It was all nothing and a man was nothing, too."  So, in order to keep from contemplating this "nada," the men prefer to stay up late in a clean well-lighted place that at least temporarily keeps the nothingness at bay.  It is in the darkness that the old man and the older waiter lie alone and sense this "nada."  But, the young man has his wife, another being, with whom to keep the nothingness from entering his thoughts.  And, since he is not alone in the night, the young man does not concern himself with the thoughts of the old man.  Instead, he wishes the old man would simply get out of his way and leave the cafe, even die:

"I'm sleepy now.I never get into bed before three o'clock. He should have killed himself last week."

Because the young waiter has "everything" as the older waiter tells him, he has meaning in his life, unlike the older waiter who, like the old man, is alone and has only "nada," seeking a clean well-lighted place to keep out the thoughts of the nothingness of existence. 

It was a nothing that he knew too well. It was all a nothing and a man was a nothing too. It was only that and light was all it needed and a certain cleanness and order. Some lived in it and never felt it but he knew it all was nada....