There is a difference. Here are 5 versions of your formula, with the normal spacing, and adding \,, \:, \; and \. In my opinion, the only places that might justify adding some space are just between the quantifiers. For the parentheses and the binary symbol, spacing is fine:
\documentclass{article}
\usepackage{mathtools} %
\begin{document}
\begin{gather*}
G\Bigl(\forall (x)\forall (y)\bigl(p(x) \vee\neg p(y)\bigr)\Bigr)\\
G\,\Bigl(\forall (x)\,\forall (y)\,\bigl(p(x)\,\vee\,\neg p(y)\bigr)\Bigr)\\
G\:\Bigl(\forall (x)\: \forall (y)\: \bigl(p(x)\: \vee\:\neg p(y)\bigr)\Bigr)\\
G\;\Bigl(\forall (x)\;\forall (y)\;\bigl(p(x)\;\vee\;\neg p(y)\bigr)\Bigr)\\
G\ \Bigl(\forall (x)\ \forall (y)\ \bigl(p(x)\ \vee\ \neg p(y)\bigr)\Bigr)\\
\end{gather*}
\end{document}
