Contrary to the common belief, a pronoun can replace a whole noun phrase, instead of a noun.
And that is why the following phrase is incorrect -
A new shirt (a noun phrase, with shirt it's head) -> A new it (Incorrect)
In your sentence we need something to replace the climate in the climate of Seoul. The noun phrase is the climate of Seoul. We can use a pronoun to replace the total noun phrase, but not a noun inside a noun phrase. So to solve this problem we need to use another class of words. We call them Pro-form.
The word that can be used as a Pro-form here, but the word it can't. That's why it's incorrect to say - it of Seoul.
The only correct version is that of Seoul.
The climate of Busan is milder than that of Seoul