What is the difference between these two persistent notifications? I use two apps which require a persistent notification: (1) the Pebble Smartwatch app, and (2) Tasker. When I pull down the notification bar (see screenshot), I see both notifications.  However, only the Tasker one shows an icon when the notification bar is hidden, and likewise only Tasker shows a slot on my lock screen.

What is the difference between how these apps are showing their persistent notification?  From my understanding, if an app wants to run continuously, it MUST put a persistent notification on the notification bar. But somehow the Pebble app is not requiring an icon or displaying on my lock screen.
These behaviors can be controlled when an app builds a notification.


  From my understanding, if an app wants to run continuously, it MUST put a persistent notification on the notification bar


You're pretty much correct until the "notification bar" part. Both of notifications are indeed persistent (i.e. on-going, cannot be removed manually by user). Take note that there isn't a button on notification drawer for clearing all non-persistent notifications.

The difference, however, is the notification's priority. This setting is independent of whether the notification is persistent or not. Usually, persistent notifications are set at least to default priority to "keep users informed about an ongoing process in the background". However, if a notification is set to minimum priority, it won't appear on notification bar.


  Priority: MIN
  
  Use for contextual or background information such as weather information or contextual location information. Minimum-priority notifications do not appear in the status bar. The user discovers them on expanding the notification shade.


Another example is Google Now's notification for weather &amp; traffic, but it isn't persistent.