† Google Notifier is Google's application for notifying you on new Gmail messages or upcoming Google Calendar events.
‡ Growl is a way of customizing the notifications to taste.
First quit Google Notifier from its menu. Then remove the file GoogleGrowl.plugin from the folder located at your home folder → Library → Application Support → Google Notifier.
Google+Growl makes Google Notifier send out Growl notifications, but for a number of reasons, it is unable to automatically disable Google Notifier's own notifications.
Google Notifier on its own is capable of show one kind of notification that acts and looks in one way. If you'd like for this notification to look differently, you have to make Google Notifier show another kind of notification that looks the way you want it to.
Growl is a way for notifications to look and feel the way you want them to. Google+Growl is a plugin for Google Notifier, and when installed works inside of Google Notifier to make it show Growl notifications as well. Using the Growl preference pane and the many available "display styles", you can then choose a look that suits you better. As an extra bonus, you can configure Google+Growl to make the contents of the notification different as well.