For example:
* When travelling, you could use a mobile app to receive instant notification of every payment made, which helps you manage your fraud risk when abroad (temporarily block the card if you see something suspect, whilst allowing you to whitelist for specific transactions so your card actually still works).
* You could have control over the "routing" of your finances... designate payments as bills and have them come out of a different account than the one you use for disposable income.
* You could split payments, go for a meal and one person pays and then agree to split and the payments are auto-managed amongst those who were there (similar to Ubers split fare)
* You could have better budgeting and management of finances, better reporting, as the banks could run the equivalent of "Google ID" (Google have this for phone number recognition) for payments so that transactions are automatically tagged and managed.
* You could have dynamic control of spending limits, set alerts and controls that follow your normal pattern to really increase fraud prevention measures, whilst your app would allow you instant control when you need to make purchases outside of that pattern.
Really it's about putting consumers in control of their money in all ways.
Banks are horribly slow, and the slowness removes all control.
Edit: I should add, there is a driving force behind this in Europe: Contactless card payments.
In Europe we've had contactless "touch" payments for several years, they are ubiquitous and can be used for any payment below £30. They are now so common that most people I know carry no cash at all, they simply touch their debit cards and the transaction is done in seconds.
This has led to very noisy bank accounts, where a bank account used to have fewer than 50 transactions per month: bills + cash withdrawals, the bank accounts now have almost every transaction: a stick of chewing gum, a sandwich from Pret, a coffee in the morning.
The greater the noise, the more speed and control are real benefits to the consumer.