Just two examples: removing Google apps from Android devices, installing non-store apps on iOS.
If you run a web search along the lines of "best apps for rooted Android" or "best apps for jailbroken iPhone" you will find many more examples of useful things that you can do with rooted/jailbroken devices.
I have a secondary Android device just out of necessity, I am a power user with enough past experience (I was active in Openmoko community, made Debian chroot for Kindle Paperwhite, am still using Nokia N900 etc.) and no - I have no actual interest in Android. Stop-gaps like Debian chroots or layers on top of libhybris won't replace the real baremetal, mainline thing.