what is the difference between \twocolumn and \documentclass[twocolumn]{book}

If you specify the document class option twocolumn, and if use the book document class, a default font size of 10pt, and paper size of A4 or USLetter, the entire document will be typeset in two-column mode, and also:

• \sloppy will be in effect,
• the default value of \parindent will be 1em,
• \textwidth will be set to either 469pt (USLetter, ca. 6.49") or 452pt (A4paper, ca. 6.25").

(The widths of the margins will depend on whether oneside or twoside is in effect.)

If you do not specify this document class option but issue the instruction \twocolumn, either in the preamble or somewhere in the body of the document, the subsequent material will be typeset in two-column mode (with a page break inserted first, if necessary), but

• \sloppy will not be in effect by default,
• the default value of \parindent will be 15pt, and
• \textwidth will be set to 345pt (ca. 4.77").

Oh, and the distance to the margin block is also affected by the presence or absence of the twocolumn option.

Of course, default page parameters can be overridden, e.g., via the macros and options of the geometry package.

Beside that some format defaults changed (as explained in Mico's answer), it is worth to note that:

• The document class twocolumn option can be used only once, but you switch between \twocolumn and \onecolumn as you want, on a per-page basis (i.e., you cannot switch in the middle of a page, this always cause a page break).

• Nonetheless, you can use \twocolumn[text in one colum] text in two columns to switch from one to two columns in the same page, without using the multicol package. Note that you cannot include some commands as \chapter{} of the standard book class in the optional argument, but it works in scrbook or memoirclass with the article option.

Example:

\documentclass[a4paper]{scrbook}
\usepackage[margin=1in,columnsep=.8cm]{geometry}
\usepackage{lipsum,microtype}
\begin{document}
\twocolumn[\chapter{Nice chapter with long title and summary}
\hrule\bigskip {\lipsum[1]}\bigskip\hrule\bigskip]
\lipsum[2-6]
\end{document}