The is used to refer to specific or particular nouns; a is used to modify non-specific or non-particular nouns.
the = definite article
a = indefinite article
For example, if I say, "Let's read the book," I mean a specific book. If I say, "Let's read a book," I mean any book rather than a specific book.
Another example, "I went to a forest. *The forest was big.". In this first forest was indefinite one, and after I said the first sentence it became definite one and used the instead.
And in your example, winter is indefinite on "that was a winter I"ll never forget." and become definite and used the in "that was the winter we went to Norway".
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