Is the set of programs that compute some function other than $h$ recursively enumerable?
The set $S$ is in fact neither c.e. nor co-c.e. You can prove this via an m-reduction to $\overline{TOT} = \{ x | \varphi_x \mbox{ is not total } \}$ (which is also neither c.e. nor co-c.e.).

Define:

$ \varphi_{f(x)}(y)=\begin{cases}
    h(y) &amp; \mbox{if $\forall z\leq y~\varphi_x(y)$ halts}\\
    undefined &amp; \mbox{ow.}
    \end{cases}
    $

Where $f$ is from the s-m-n theorem.

Then you have that $\overline{TOT} \leq_m S$ via $f$ because, for all $x$:

$x \in TOT \iff \forall y~\varphi_x(y) \mbox{ is defined} \iff \forall y~\forall z\leq y~\varphi_x(z) \mbox{ is defined} \iff \forall y~\varphi_{f(x)}(y)=h(y) \iff f(x) \notin S$