The first line of this saying is quoted in 2 Peter 2.22.

“Like a dog that returns to his vomit”: It is known everywhere that after vomiting a dog will lap up its vomit. So “returns to his vomit” means “eats its vomit again.”

“Is a fool that repeats his folly”: The “fool” does not learn from his foolish errors but rather repeats the same foolishness over and over. The comparison is between the dog repeating the eating of its vomit and the fool repeating his mistakes. In translation it may be necessary to say, for example, “Fools always repeat their mistakes like dogs always eat their vomit.” See Good News Translation.

Quoted with permission from Reyburn, William D. and Fry, Euan McG. A Handbook on Proverbs. (UBS Helps for Translators). New York: UBS, 2000. For this and other handbooks for translators see here .