Not a thing—the words are synonymous.
Ornithologists tend to use the word “bill” more often than “beak.” Some people use “beak” when referring to songbirds with pointed bills, and “bill” when discussing birds like ducks with more fleshy beaks. However, both words are used in reference to a wide variety of species.
One important caveat: if you tell that old joke about the duck going into the drugstore to buy some chapstick, and the cashier asks how he’s going to pay for it and the duck says, “put it on my bill”—well, that joke just doesn’t work if you say “beak.”
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