日本は南北に長い国なので、南と北では気候が大きく違い、…
(From Tobira Gateway to Advanced Japanese, p.5.)
I thought 大きく is an adverb, so it can only qualify a verb; but 違い is a noun. What's going on here?
This 違い is not a noun but a verb, and that's why it's modified by an adverb (大きく). This 違い is the continuative form (連用形) of 違う.
The continuative form of a verb has various usages. You can read a summary here: https://japanese.stackexchange.com/a/65953/5010 The second and the fifth usages in the last list are relevant to you now.
違い here is not a noun, but 連用形 of the verb 違う.
Generally it is called 連用終止 in Japanese grammar (for Japanese people, at least).
Also note 連用形 is often used as a noun like 違い (連用形の名詞化 or 連用形名詞(pdf)), hence the confusion.
Using 違い sounds more for writing than 違って.