How many orders of magnitude does $345,632$ differ from $567,123,423$? If you just divide you get: $$\frac{567123423}{345632}=1641(nwn)$$
This number is closer to $10^3$ than $10^4$ so I would have answered $3$ orders of magnitude.
But the book says to first express the numbers in scientific notation and then round the decimal number like so:
$$\frac{567123423}{345632}=\frac{5.7*10^8}{3.5*10^5}=\frac{10*10^8}{1.0*10^5}=\frac{10^9}{10^5}=10^4$$
The book gives an answer of 4 orders of magnitude.
What is the correct answer here?

Both, I'd say. Orders of magnitude is a fuzzy subject. Although your approach is more correct, as in closer to the unrounded answer, the book's approach is a lot easier to do (I can't divide by $345\,632$ in my head), and in the end isn't that one of the reasons to use a concept like "orders of magnitude" in the first place?
That being said, $\frac{5.7}{3.5}$ is clearly seen to be between $1.5$ and $2$, so in the end I would prefer to call it three orders of magnitute, personally.
