These are two ways of saying the same word. Often we spell the first one with an apostrophe – ‘possum – to indicate the missing ‘o’. Americans usually say ‘possum, because it is easier to say in fast casual speech, and Americans use the word a lot, because ‘possums are everywhere in America! Under your house, in your garbage, in the woods, in the city, etc.
Opossum is awkward to say in fast speech because it does not have the sound of an English word; it was, originally, a Native American (‘Indian’) word. Opossum is said mainly in formal contexts, or by non-American English speakers.
In case you don’t know . . . an opossum is an animal that looks like a giant white rat, which lives almost everywhere in America. Most people don’t like them (the poor creatures), probably because they look like giant rats. Also, they may bite pets and carry disease.
The Popular Phrase "Playing Possum"
The phrase "playing possum" originates from the behavior of the opossum, which play dead when threatened by predators.
In a human context, the phrase can be used to denote an act of feigning death or unconsciousness to ward off a human attacker, or it can suggest that a person is pretending not to understand or know something when they do in fact know it very well. A person might "play possum" in order to avoid conflict, being verbally attacked, or called upon to perform an undesirable duty.