How do I "pin" Notes in Mountain Lion?
The feature description is a bit misleading. It appears that by "close Notes", Apple means "close the main Notes window", not "close the Notes app", which (not surprisingly) leaves other windows open.

In fact, the context menu refers to the main Notes window as Notes:



So what Apple's description basically means:


  Even if you close Notes, the note stays open on your desktop.


= You don't need to use the overview window at all times. You can use Notes.app similar to Stickies.app

By the way, Stickies.app is still available in Mountain Lion and has been upgraded to version 9.0.