Re: Well even though my bank has an app
Here's why I use it. I don't know about the generational thing; I'm in my late 30s, but I'm also an Android developer.
First, I can check my balance instantly on the login screen without having to log into the app (I also use the fingerprint reader on the phone so if I lose it, no one can access that information unless they have my finger, or, if they reboot the phone, the phone's pin). I use my cards a lot, far more than cash, so knowing how much is in there before I go to the checkout is really important.
Second, if I do want to make an impulse purchase, then I can transfer between accounts right there and then. In addition, if I want to send someone money I can do it right there and then. I could even - although I don't do this - bump phones for the NFC high five.
Third, the UX is far better than the website. They put a lot of work into the UX, and when it's in an app it's not browser dependent (so Chrome, Firefox, Opera, etc. which could all render in different ways are replaced by a uniform app).
Fourth, my sixteen character website password doesn't have to be typed in, because the device has a pin (up to six digits), and I don't need to enter my username/customer number. All I need to know and type in is my pin.
Fifth part one, it's far more secure than the browser - any browser. There's extra layers of security you can't have in the browser, even on the computer. The app doesn't depend on the security of other applications (like the browser, or plugins for the browser). It's sandboxed (and all others should be too) so no other applications can open it, and it can only connect to one site. It's never completely secure because you could still bypass it with (serious) user intervention, but it's far more secure than the browser.
Fifth part two, it's linked to my phone. So only that device can use it, and all devices have to be registered with online banking in the first place. My bank gives us a maximum of three devices, and if anything changes substantially in the device then the app needs to re-register. So you can only use it on that phone; you can only use new devices after you've done a couple of 2FA routines, so people can't just register new devices without having access to your online banking, your email and your phone.
tl;dr it's really fast, convenient, makes managing my money so much easier, looks pretty and more secure.