From one to another or From one to the other?

Is there a difference if I say "the recipe varies from one cook to the other" or "the recipe varies from one cook to another"?

The difference between "the other" and "another" in the phrases "from one cook to the other" and "from one cook to another" is basically the same one that distinguishes "the X" from "a [or an] X": The phrase "from one cook to the other" implies that the speaker or writer has already introduced two specific cooks to the hearer or reader and is now referring to those specific persons. The phrase "from one cook to another" refers to multiple cooks (at least two) who are not more specifically identified for purposes of the immediate action described.

As RegDwigнt's comment notes, the most common circumstance for the phrase "from one cook to another" involves "an unknown number of cooks we don't know." However, the wording can appear in situations where we know both the number and the identity of the cooks. For example:

Five fry cooks—Sid at The Corner Kitchen, Niko at Spiro's, Ray at Gary's Grubland, Tina at Silvertone Diner, and Baltasar at Le Fud—are famous in Traylorville for their signature Liver & Onions platters, though the recipe varies from one cook to another.

The reason you wouldn't use "the other" in place of "another" in this example is that more than two cooks are involved. In a situation involving only Sid and Baltasar, for example, a hearer or reader would assign one of them the role of "one cook" and the other the role of "the other." But in a situation involving five cooks, this isn't possible. In effect, "the recipe varies from one cook to another" constitutes an alternative way of saying "each of the cooks uses a different recipe."