Sparrow iPhone and Push
The difference between the Mac and the iPhone is the Mac version is awake at all times. On the iPhone, iOS systematically suspends all apps activity after 10 minutes maximum making it impossible to send you notifications.
Apple has an API for apps to be notified in the background, but currently it’s only allowed for VoIP. So it’s not possible to build an iOS mail client that’s on par with the built-in one.
[...] schedule. Meanwhile, the built-in Mail app can check for new messages on a repeating schedule, but third-party apps cannot. See also the background music [...]