They are the same thing but Northerners tend to use quyển more while Southerners prefer cuốn. The former is the standard Sino-Vietnamese reading of 卷 while the latter is a further Vietnamisation of it.
One is used in the north. The other is used in the south. I don't remember which used in which side. But they are the same thing.
Yes, quyển sách is 卷冊 and 卷 also gave rise to cuốn. In Modern Vietnamese "冊" (sách) means a "book" while "書/书" (thư) means "mail/letter". A library in Modern Vietnamese is not "圖書館／图书馆" (đồ thư quán) but "書院/书院" (thư viện).
both "cái" and "quyển/cuốn" are classifiers. while "cái" is a general classifier for objects, "quyển/cuốn" are specific classifiers for books.