Strutting around: What's the difference between \strut, \mathstrut and \vphantom?

Each are defined as follows in the default document classes (article, book and report):

\strut

\def\strut{\relax\ifmmode\copy\strutbox\else\unhcopy\strutbox\fi}

with \strutbox defined as

\setbox\strutbox\hbox{%
  \vrule\@height.7\baselineskip
        \@depth.3\baselineskip
        \@width\z@}%

\mathstrut

\def\mathstrut{\vphantom(}

Using \mathstrut would be equivalent to using \vphantom(, with \strut sets a zero-width rule (\@width\z@) with depth .3\baselineskip (\@depth.3\baselineskip) and height .7\baselineskip (\@height.7\baselineskip). Note that the latter is font-dependent, since it uses \baselineskip. That is, it is modified with font change selections using \fontsize{..}{..}\selectfont, including the use font switches like \small, \large, etc.

I would use \strut within text or math, and use \vphantom if there is anything else I want a specific height of without the horizontal displacement. For example, when breaking two lines of math with different heights but still wanting to use extensible delimiters \left and \right. \mathstrut is specific to the size of ( and (quoting barbarabeeton), "\mathstrut is often better in math than \strut. Which one is better depends on the local context."

Loading the amsmath redefines these boxes somewhat, although their interpretation remains the same.

The \strut and \mathstrut commands just define a box with a zero width and a convenient fixed height and depth. The last command is for math mode.

The \vphantom commands lets you create a box with a zero width and the same depth and height as the material that's in the argument of the command. The \vphantom command is ideal for scaling left and right delimiters, as is explained in the following example, which is from LaTeX and Friends. The example requires amsmath. The other commands aren't as flexible.

\begin{align*}
f & = g\left( 3^{3^{3}} + \text{\textrm{\ldots}}\right.\\
  & \qquad \left.+ 3\vphantom{3^{3^{3}}} \right)\,.
\end{align*}