This blog came to me in a dream last night. Let's do science.
Opossum: Critter
Gopher: Varmint
Groundhog: Critter
Fox: Varmint
Armadillo: Critter
Raccoon: Both (situation-dependent)
Squirrel: Critter
Cat: Neither
Skunk: Critter
Dog: Neither
Wolf: Neither
Coyote: Varmint
Chipmunk: Critter
Prairie dog: Varmint
Feral hog: Varmint
Beaver: Varmint
Porcupine: Critter
Badger: Varmint
Bunny: Neither
Jackrabbit: Varmint
Rat: Vermin
Pet rat: Critter
Hallway food goblins: Varmints
It becomes clear, after rigorous fieldwork, that the critter/varmint binary is actually more of a spectrum. Critter is the baseline for an animal that's small, but not too small. Varmint implies some level of "pesthood"—eating crops or destroying property or whatnot. A critter becomes a varmint when it takes your stuff. But this is not an absolute! It depends on the stuff. A raccoon vacillates between critter and varmint depending on what particular trouble he is up to at the moment. Whereas the opossum simply wants your garbage, you are happy to let him have it and thus he remains firmly ensconced in critterdom. He is perhaps the apex critter. The gopher, on the other hand, is the Platonic ideal of a varmint: he digs up your yard and you must perform some element of labor, or hire professionals, to get rid of him. Critters come and go; varmints must be addressed. We land upon a fundamental theory of critter vs. varmint: A varmint is a critter where you don't want him to be.