My basic issue with the up button is the way you describe it conflates it with the back button, while the two should be considered different things (even though in many cases they will do the same thing). I blame Google more than anyone for this as they themselves conflate the issue mightily by, for example, styling the "up button" so it looks exactly like a back button.

How would you detail the differences between the 2 buttons?

Lets assume for a moment you are using the browser on your phone when a notification pops up alerting you to a new email. Clicking on the notification brings up your mail app and displays the email you just recieved.

At this point if you click the back button you will be returned to the browser session you were previously viewing. If, however, you were to click the up button instead you would remain inside your mail app and go “up” from your message to the inbox.

Just to tack on my simple opinion: 'Up' is in context to the current application. 'Back' is in context the users perspective.

If I'm in the browser and click a mailto link, it opens the email client. The 'up' metaphor should return me to the inbox. The 'back' metaphor would return me to the web browser/webpage. (Did not realize I was using the exact same metaphor as my sibling, sorry)