Change behaviour of a custom command

I don't think that you need to define a new command. The following code will do the job.

begin{itemize}
    item [$ast$] Point 1
    item [$bullet$] Point 2
end{itemize}

Write the symbol you want to use for label/bullet type in square brackets after item. It will do the job.

You can make your list-like construction CVItemListStart...CVItemListEnd act like enumitem's label-controlled environment in the following way:

documentclass{article}

usepackage{enumitem}

newcommand{CVItem}[2][defaultbullet]{%
  item[#1] {small #2par}
}

newcommand{defaultbullet}{textbullet}
newcommand{CVItemListStart}[1][textbullet]{%
  begingroup
  renewcommand{defaultbullet}{#1}%
  begin{itemize}[label=#1]}
newcommand{CVItemListEnd}{%
  end{itemize}
  endgroup
}

begin{document}

First list:
CVItemListStart
  CVItem{First}
  CVItem[$ast$]{Second}
  CVItem{Third}
  CVItem[--]{Final}
CVItemListEnd

Second list:
CVItemListStart[$ast$]
  CVItem{First}
  CVItem{Second}
  CVItem{Third}
  CVItem{Final}
CVItemListEnd

Third list:
CVItemListStart
  CVItem{First}
  CVItem{Second}
  CVItem{Third}
  CVItem{Final}
CVItemListEnd

end{document}