Differences between \stackrel and \stackbin

Don't use \stackrel or \stackbin, they are obsolete. Use instead \underset and \overset from amsmath as they automatically space things correctly (whether it's a binary relation or a binary operator):

\documentclass{article}

\usepackage{amsmath}

\begin{document}

$f(x) \overset{\text{def}}{=} x \ln(1+x)$

$f(x) \underset{x \to 0}{=} x^2 + o(x^2)$

$A \underset{\text{below}}{\overset{\text{above}}{+}} C$

\end{document}

Without the stackrel package, \stackrel is defined in ltxmath.dtx as

\def\stackrel#1#2{\mathrel{\mathop{#2}\limits^{#1}}}

which typesets a relational operator with a top limit (effectively placing it on top). Heiko Oberdiek's stackrel package is said to provide an

Enhancement to the \stackrel command.

This "enhancement" provides an optional argument to \stackrel for placing something below the relational operator (using a similar process as the original \stackrel). Additionally, it provides a counterpart for binary relations called \stackbin. The difference between the two (or when to use which one) is contained within the post What is the difference between \mathbin vs. \mathrel? Here is a similar take on the use of stackrel.

\documentclass{article}

\usepackage{stackrel}

\begin{document}

\begin{tabular}{clc}
  \multicolumn{3}{c}{Relations} \\[5pt]
  \LaTeX & Typeset & width \\
  \hline
  \verb|$x=x$| & $x=x$ & \setbox0=\hbox{$x=x$} \the\wd0 \\
  \verb|$x\stackbin[c]{a}{=}x$| & $x\stackbin[c]{a}{=}x$ & 
    \setbox0=\hbox{$x\stackbin[c]{a}{=}x$} \the\wd0 \\
  \verb|$x\stackrel[c]{a}{=}x$| & $x\stackrel[c]{a}{=}x$ & 
    \setbox0=\hbox{$x\stackrel[c]{a}{=}x$} \the\wd0 \\[10pt]
  \multicolumn{3}{c}{Binary operators} \\[5pt]
  \LaTeX & Typeset & width \\
  \hline
  \verb|$x+x$| & $x+x$ & \setbox0=\hbox{$x+x$} \the\wd0 \\
  \verb|$x\stackbin[c]{a}{+}x$| & $x\stackbin[c]{a}{+}x$ & 
    \setbox0=\hbox{$x\stackbin[c]{a}{+}x$} \the\wd0 \\
  \verb|$x\stackrel[c]{a}{+}x$| & $x\stackrel[c]{a}{+}x$ & 
    \setbox0=\hbox{$x\stackrel[c]{a}{+}x$} \the\wd0
\end{tabular}

\end{document}

Note the equivalent spacing using \stackrel with =, while similar spacing is returned using \stackbin and +. In essence, use \stackrel for relational operators, and \stackbin for binary operators.

I personally prefer using \mathop, because it's simpler. Instead of using nested under/oversets, you could simply write:

$A \mathop{+}_{\text{below}}^{\text{above}} C$