For me, a physical note occupies space on my desk but rarely ever gets in the way. I can look at it if I want but it is not annoying like notifications in cell phone or pop ups in desktop apps. This includes both, sticky notes and notepad.
Also another part of using pen and paper is drawing capabilities. Sometimes I have ideas that will take too long to describe in words, but one crappy flowchart grabs everything about the idea.
And delete function in real world is so hard to replicate in electronic devices. When you are done with a physical note, crumbling the paper and tossing in trash is so satisfying.
With all that said, all my physical notes are temporary. They are either to do lists which get done in next few days or ideas that get transferred to more permanent place such as journal or blog post.
That seems to be an actual problem for everyone, isn't it?
Maybe a notes app could try solving it by doing the Facebook-like thing where it occasionally shows you a random note from the past. :)
Or suggesting similar past notes when you write a new one that matches them. e.g., if I'm writing down an idea I've had for a role-playing game, the app suggests past notes with RPG in them.