The main difference between a rat and an opossum is that the rat is a rodent, and the opossum is a marsupial. This means that the opossum has a pouch that protects its babies. Though both animals have adapted to living around and among humans, the rat has become especially adept at living in human habitations, including ships, sewers and subways.
This close relationship between rats and humans can be destructive in ways that are not seen with the opossum. For example, rats are believed to be the vector for devastating plagues and other diseases. Great numbers of rats not only eat crops but befoul them and render them useless.
The opossum not only has a pouch, but both sexes have bifurcated reproductive organs. The placenta of the opossum is primitive and doesn't really function. Though baby rats are helpless when they're born, baby opossums are nearly in an embryonic state. They can just barely crawl into their mother's pouch and latch on to a teat there. This reproductive physiology is very different from a rat's.
Rats are also far more widespread than opossums, who are not found in climates that are too harsh. By contrast, the brown and black rat are found everywhere save Antarctica.