Installed App with Brew Cask Over Existing Install, Did it Remove the Old App Files?

Cask does not clean up old versions by default. It stages the installs in /opt/homebrew-cask/Caskroom/$APPNAME and then symlinks them in to ~/Applications or /Applications depending on the brew recipe. And then it leaves the old versions around until you run brew cleanup.

You can see old installs of OpenOffice under /opt/homebrew-cask/Caskroom/openoffice. You can safely delete the old installs from there, though brew cleanup is supposed to do this for you.

If you had previously installed OpenOffice using their DMG installer and not Homebrew Cask, it was most likely installed in /Applications and not ~/Applications. Cask won't install over top of an existing installation that wasn't previously Cask-installed itself. So check /Applications and make sure you remove the old version from there if hadn't installed it via Cask. If you had installed it via Cask all you need to do is delete it from /opt/homebrew-cask/Caskroom/openoffice as mentioned above.