\usepackage{} -- graphicx vs. graphics vs. psfig/epsfig

Using \usepackage{graphicx} is preferable to \usepackage{graphics} because although "both packages have the same capabilities, ... the resulting syntax may be more cumbersome [with the graphics package]."  TUGboat, Volume 17 (1996), No. 1, "Tutorials" (p. 43) -- PDF,

With \usepackage{graphicx}, "you can incorporate graphics in pdf, jpg, png or tif (with one f) formats. If you don't specify the file extension, it will look for and use whichever one is available."  However you may need \usepackage{graphics} if you have .eps graphics.

% \usepackage{psfig} and \usepackage{epsfig} are obsolete. (Link here.)
% \usepackage{epsf} % Obsolete, low-level typesetting of Encapsulated PostScript (EPS) files.

\begin{figure} \centering vs. \begin{center}

\centering doesn't leave vertical space before and after it. 

\begin{center} is an environment.  Also, the \centering command does not start a new paragraph.

Using \begin{figure} \centering is thus preferable to \begin{center}.

\emph{emphasized text} vs. \textit{italicized text}

\emph{emphasized text} vs. \textit{italicized text} -- If your captions are already italicized, \textit won't show any effect.  \emph can be nested such that an emphasized phrase within an italicized phrase will look like this. 
_NEED REF_

\captionof{element}{Caption text.} vs \caption{Caption text.}

\captionof{figure}{Caption text.} vs \caption{Caption text.} ... _NEED REF_ These have different purposes/effects. 
\captionof{table} or other elements work in this manner.