What's the difference between "another" and "other"? [duplicate]

There's a formula:

another = an + other.

Think of it as of an article plus the word "other" that have historically merged into one word. Grammar requires some article before "other book"; either "the" or "a." Depending on the context, you get either "You need to buy the other book" (if, for instance, the guy bought only the first book out of the set of two) or "You need to buy an_other book" (any other).

Another implies extra or additional with respect to the current amount. "I'm going back for another sandwich."

Other is an adjective implying it is different than the object in discussion. "Pass me my book. Not that one, the other one." It can also be a pronoun referring to things or people: "Let's wait for the others to arrive."

Going back to your example, if I said "You need to buy the other book," I'd be implying a mistake has been made, not that you should buy an additional book as if I had said "You need to buy another book."