What is the difference between \em and \emph?

\emph is like e.g. \textit a command with an argument. \em is the "switch" variant, comparable to \itshape. \em is not an outdated TeX or LaTeX2.09 command but a real LaTeX2e command. Actually \emph is defined through em:

\DeclareTextFontCommand{\emph}{\em}

\em is useful for long texts (\emph e.g. doesn't allow the argument to contain a \par). The commands differ (like the \textit/\itshape) in their handling of the italic correction:

\documentclass{article}
\begin{document}

abc\emph{lll}lll  \textit{lll}llll

abc{\em lll}lll  {\itshape lll}llll

abc{\em lll\/}lll

%\emph{abc\par bc} error

{\em abc\par bc}
\end{document}

l2tabu (a guide to obsolete commands and packages, available in English and, more up-to-date, in German) comments on \em as follows:

May be useful when defining macros. In continuous text \emph{...} should be preferred to \em.

(Table 1, p. 8 [en] / 10 [de])

\emph is a LaTeX2e command and \em is a LaTeX 2.09 declaration, see e.g. http://www.public.asu.edu/~rjansen/latexdoc/emph.html . Since you're probably using LaTeX2e you should use \emph.

• \emph is a command and is used like \emph{text}.
• \em is a declaration and is used like \em text or {\em text}.

Their differences may be demonstrated by the following:

\documentclass{article}

\begin{document}

some \em text% Works as intended

some {\em text}% Works as intended

some \em{text}% Does not work as intended, "text" is not emphasized

some \emph{text}% Works as intended

some \emph text% Does not work as intended, only "t" of "text" is emphasized

\end{document}