I have some bills which will get "lost" if they are not sent at least certified mail. Sent electronically, it isn't anywhere near as concrete proof  as a piece of physical mail sent with a signature trail.
Paper complaints, especially legal work are hard to ignore. E-mail, even calls, there is no paper trail and can be hidden. However, a certified message either gets received or it gets refused. Either way, someone had to interact with the document in a provable way. Even now, our society isn't paperless and when it comes to legalities, there is no beating physical paper documents.
As for banking software on my phone, I'd say that iOS and Android have a better record for security with a few caveats , than a desktop machine. However, my biggest concern with too much stuff on a phone is if it gets stolen. Of course, the ideal would be having the banking software use KeyChain or Google's equivalent and ask for a PIN before it is run, so all sensitive data is encrypted, not just by the phone's encryption, but by a specific API.
: Proof to judges and juries who are still in the pen and paper age, and those are the people who need to be persuaded if worse comes to worst. You can talk crypto all day long to a jury, and their eyes will glaze over. Show them a piece of paper with a physical John Hancock on it, they will make a decision.
: Depending on how well the jailbreak is crafted, it can destroy's iOS's security, where an app can go hog wild and do what it wants to.
Android is more secure in this regard, but apps ask for a lot of permissions . However, a rooted Android device is just as secure (barring a dumb user) than a non rooted one. Attacks like a compromised Web browser will have far less effect than they do on a PC.