I wonder if apartment building would technically refer to a more modern building specifically built for apartments as opposed to the palazzi which today are being "used as or converted into" apartments? Palazzi were originally built as large homes for the wealthy and powerful families but today have become apartments in that they house many people or they are used as museums (I saw lots of this in Venice).
An important "palace"? Seriously. That's redundant, if you're dealing with palaces [coming from someone who doesn't :D]. I think in this case, the better translation would be "building" which I saw as a hint. I used it and it was still marked as correct, but I think that palace would be a lesser used translation here.
In American English (at least in my experience), the word "palace" refers (in a literal sense) only to a place where kings and queens live. Sometimes large houses are called "palaces" but only in an ironic/over-the-top complimentary way: "This place is [like] a palace." We refer to very large private houses as mansions. And tacky, overly ostentatious, architecturally incongruous mansions are called "McMansions."