Both typographical variants of the symbol are in wide use. There is no difference in meaning, and readers can be expected to recognize either as a symbol for the empty set. Many people have quite distinct (and different) opinions about which one looks "best" or "right", though.

It's up to personal preference, really.

(The fact that TeX names one of them \emptyset and the other \varnothing, suggesting it is a "variant" symbol, merely reflects the preference of TeX's author Donald Knuth for the form, not a collective decision by any group of mathematicians).

Try \varnothing from the amssymb package. It is perfectly round, and the comprehensive LaTeX symbol list states that it is preferred by many to \emptyset.

\documentclass{article}

\usepackage{amssymb}

\let\oldemptyset\emptyset
\let\emptyset\varnothing

\begin{document}
$\oldemptyset$ $\emptyset$
\end{document}

According to Wikipedia André Weil (and maybe others) introduced the symbol $\emptyset$ for the empty set. They were inspired by the letter Ø in the Norwegian and Danish alphabet. So, I guess you could argue that the symbol should be pronounced as it is pronounced in these alphabets.

In my experience most mathematicians just call the symbol "empty set".

Note also, that the Wikipedia article says that the symbol is not to be confused with the Greek letter $\phi$ or $\Phi$.