Using your code snippet as an example, \centering only has an impact on the \caption. However,
\hspace*{\fill}
{ILLUSTRATION 1} \hfill {ILLUSTRATION 2}
\hspace*{\fill}
is meant to set ILLUSTRATION 1 and ILLUSTRATION 2 even distributed across \textwidth:

\documentclass{article}
\usepackage{showframe}% http://ctan.org/pkg/showframe
\begin{document}
\begin{figure}
\centering
\hspace*{\fill}
{ILLUSTRATION 1} \hfill {ILLUSTRATION 2}
\hspace*{\fill}
\caption{Example float with two illustrations}
\label{fig:mult1}
\end{figure}
\end{document}
However, replacing \hspace*{\fill} with \hfill, the space from the right margin is "gobbled":

\documentclass{article}
\usepackage{showframe}% http://ctan.org/pkg/showframe
\begin{document}
\begin{figure}
\centering
\hfill%\hspace*{\fill}
{ILLUSTRATION 1} \hfill {ILLUSTRATION 2}%
\hfill%\hspace*{\fill}
\caption{Example float with two illustrations}
\label{fig:mult1}
\end{figure}
\end{document}
giving value to the command definition for \hspace*:
LaTeX removes horizontal space that comes at the end of a line. If you don't want LaTeX to remove this space, include the optional * argument. Then the space is never removed.
An alternative to use if \hfill is more intuitive than \hspace*{\fill}, is to insert dummy text (like \null or \mbox{}) at the (start and) end of the line. That way the \hfill will always have "something to infinitely stretch against":
\null\hfill%
{ILLUSTRATION 1} \hfill {ILLUSTRATION 2}%
\hfill\null%
In the above examples, showframe merely highlighted the text block.
a\hfill b\hfill \hspace{0.01mm}– Aug 11 '15 at 06:23