I've always interpreted "power user" in this sense to be someone who occasionally wants to do something specific with their device that may require specific tools.
Like a "normal" user (of a phone) will make calls, texts, browse the web, check email, maybe watch a movie or play a game. A power user may just be someone who wants a custom keyboard for some reason or even just wants to be able to download a file via the browser, navigate to it in the file system, maybe unzip it, copy it elsewhere, open it with another app, etc.
Granted these are just my personal examples because they were things I found exceedingly difficult to do on my old iPad (although I guess the keyboard thing has been added). It wasn't something I had to do all the time but something as simple as browsing the folder structure and finding/moving the file I just downloaded was gonna require me to jailbreak the thing. On my Android phones and tablet, it's not something I often need to do but when I do, it's just a matter of using a file browser.
I think "power user" implies using more basic/powerful features outside of launching apps from a set of icons. Then again, some of the awesome projects I've seen controlled via an iPad prove that iOS users can also be power users as well.