The main difference between the two futures presented in "Wall-E" and "By the Waters of Babylon" is the reason that mankind was not really on earth anymore. In "Wall-E," it is because we produced too much trash; the earth was filled with waste, rendering the atmosphere to toxic to breathe, and the earth too full to habitate. So, mankind left, awaiting the day that they could return. In "By the Waters of Babylon," mankind is almost obliterated by nuclear bombs. There are a few survivors that remain on earth and survive there, instead of leaving. The earth was filled with radiation so bad that the survivors were "forbidden to touch metal" for fear of radiation poisoning. So, that is the main difference.
Other differences lie in the way that the two surviving groups live. In "Wall-E" the survivors live in comfort and ease, surrounded by technology. They have accurate records of earth and what it was like, and how and when to return. In "By the Waters of Babylon," the people have regressed; they live in tribal, ancient conditions with no technology whatsoever. To go with it, they have rituals, rules and rites that are passed down from generation to generation, whereas in "Wall-E," the people don't really think much, or do much. They just exist, mindlessly.
Similarities exist in the endings of these stories. Both groups find a way to have hope, to return things to the way that they have been. One person (or robot) discovers hope for the continuation and progression of mankind. They are also similar in that the large masses of people live life as is, and don't challenge the norms. There is one characters (John and Wall-E) that challenges the norms and helps lead mankind to the next step in survival.
I hope that those thoughts help to get you started. I also provided a link below that gives tips on writing compare/contrast essays, and that should help too. Good luck!