First of all, \centering needs a \par (or an empty line) before the closing brace
{\centering
contents\par
}
Usually it's used inside some environment that provides the necessary \par, such as minipage or figure. Try
abc{\centering def}ghi
to see the effect.
The main difference with center is that \centering doesn't leave vertical space before and after it: \begin{center} is defined in terms of trivlist.
\def\center{\trivlist \centering\item\relax}
\def\endcenter{\endtrivlist}
\def\centering{%
\let\\\@centercr
\rightskip\@flushglue\leftskip\@flushglue
\parindent\z@\parfillskip\z@skip}
Note that it's an error to use
\center
because this will last “forever” (up to the end of the current group, anyway). The command exists just as a side effect of the existence of the center environment.