The main reason I see to use minipage over \parbox is to allow verbatim (\verb, verbatim, etc.) text inside the box (unless, of course, you also put the minipage inside a macro argument).
EDIT Here are other differences between minipage and \parbox (from the comments to Yiannis' answer and from looking at the source code of both these macros in source2e).
A first difference, as already mentioned by lockstep in his question, is in the footnote treatment: minipage handles them by putting them at the bottom of the box while footnotes are lost in a \parbox (to avoid this, you must resort to the \footnotemark/footnotetext trick):
\documentclass{article}
\begin{document}
\parbox[t]{3cm}{text\footnote{parbox footnote}}
\begin{minipage}[t]{3cm}text\footnote{minipage footnote}\end{minipage}
\end{document}
A second difference is in that minipage resets the \@listdepth counter, meaning that, inside a minipage, you don't have to worry about the list nesting level when using them. Here's an example which illustrates the point:
\documentclass{article}
\begin{document}
\begin{list}{}{}\item\begin{list}{}{}\item\begin{list}{}{}\item\begin{list}{}{}\item
\begin{list}{}{}\item\begin{list}{}{}
\item %\parbox{5cm}{\begin{list}{}{}\item \end{list}}% error
\item %\begin{minipage}{5cm}\begin{list}{}{}\item \end{list}\end{minipage}% no error
\end{list}\end{list}\end{list}\end{list}\end{list}\end{list}
\end{document}
A third difference is that minipage sets the boolean \@minipagefalse which in turn deactivates \addvspace if it's the first thing to occur inside a minipage. This means that minipage will have better spacing and allow better alignment compared to \parbox in some cases like the following (left is minipage, right is \parbox):
\documentclass{article}
\begin{document}
Pros: \begin{minipage}[t]{3cm}\begin{itemize}\item first \item second%
\end{itemize}\end{minipage}
Cons: \parbox[t]{3cm}{\begin{itemize}\item first \item second\end{itemize}}
\end{document}
You can use a minipage with the width as a proportion of textwidth.
\documentclass{article}
\usepackage{lipsum}
\begin{document}
\fbox{\begin{minipage}{0.9\textwidth}
\lipsum[1]
\end{minipage}}
\end{document}
The \begin{minipage}...\end{minipage} puts a block of text in its own block. The 0.9\textwidth provides the sizing parameter.
To enclose everything in a frame you can use \fbox.
Best Answer
The following solution patches
\@iiiparboxto get the height in the second parameter. The value is stored in the dimension register\parboxheight.If the height is not set, LaTeX uses
\relaxfor this parameter. Then,\parboxheightis set to zero. If you want to know, whether the height parameter was set the optional argument, an additional switch can be used.Example:
With switch: