Maybe regional or depending on what bank you have. Here in Sweden you can do most things from your phone, so there isn't much of a reason to go to the desktop site usually. It's also marginally more convenient to log in on the phone app, as the authentication can be done on the phone.
Bills become much easier to pay via the phone, as you can scan the information from the invoice with your camera (for those companies that refuse to make electronic invoices available). Many invoices now use QR codes as well, so you only have to scan one thing to get all the information needed.
None that I can share, but my source is the major US bank's software that I work on. More than two thirds of our traffic is handled on mobile devices now.
I don't know anyone who does banking on their phone -- it's all desktop. Granted, I haven't taken a formal survey or anything.