I switched from an iPad Mini to the current generation paperwhite.
Personally, I FAR prefer the reading experience on an eInk display. I have no strong allegiance or preference towards Kindle's eInk devices, but that said I am at this point invested in the Kindle format (I've bought a number of ebooks on amazon).
For me, it comes down to a few main points:
1) It's FAR less distracting for me to read on a kindle (I'm just going to say "kindle", but I don't mean the android) then an ipad. It's honestly refreshing for me to have something so "simple" as a book in front of me, laying in bed. I don't have to fight the urge to check my email, or refresh the redit homepage, or play some stupid video game, etc. I have only a single option: read the words on the page in front of me.
2) Even though the battery life on the iPad is awesome awesome, it's not as good as the kindle. AND honestly, the battery life on the ipad mini blows me away. But even still, it's something that I do think about. On my Kindle, I never, ever think about it. Sometimes for weeks on end, reading every night, I don't even think about it or notice it. When I do get a low battery notice, I can normally still read for another full night (at least) before actually having to plug anything in. For me, this is huge. I don't need to think about taking a charger with me on trips, etc, unless it's a very very long trip. Of course, being a nerd, I always have a micro-usb charger somewhere close (and the fact that it's a standard micro-usb cable is a huge plus), but it's a weight off of me to not have to think about it.
3) The display itself, for me, is an absolute must. I spend my entire day sitting on a computer, reading text (well, programming, which involves lots of text reading!). For me, I don't get headaches or anything looking at a display like this all day. I don't "mind" it. But, it's NOT the same as looking at a book. eInk is, obviously, closer to a paper book then a led/lcd screen. Again, the resolution on the ipad rmini is amazing, super super impressive, but for me that matters less then the "look" and feel of the display.
All of this said, if I was going to buy one or the other, and reading was only a part of what I was interested in doing, I'd certainly choose an iPad Mini over a kindle. So take all of this with a grain of salt.
p.s. I actually preferred the kindle without the backlight. I had the kindle keyboard and loved it. The backlight is super super convenient, but for me takes away a bit of the "book" feel.