I rarely use the native email clients from Android and iPhone. Gmail's mobile web app has a better interface and it's easier to use than those applications, but there are some limitations that force you to switch to the native apps: you can't attach files and there's no support for notifications. Until mobile browsers add support for these features and integrate with the operating system, native apps will continue to be more powerful.
The latest version of Google Mobile App for iPhone solves one of these issues by adding push notifications for Gmail and Google Calendar. You no longer have to use Google Sync to get notifications when you receive an email message or an event is about to occur.
"Now Google Mobile App can help with push notifications from your Google account to your iPhone -- an icon badge shows you’ve got new mail in Gmail, and Google Calendar event reminders appear right on your home screen."
After updating Google Mobile App 0.6, it seems that Google only shows a badge when you get a new message, just like the native mail client.
Update: Google's mobile blog says that "you will only receive pop-up messages and sound/vibration for calendar alerts - new email is shown using the unread count on the Google Mobile App icon and in the apps tab".
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16 hours ago