I have a Kindle and a Kobo. Actually several of each. I have a strong preference for the Kindles for a number of reasons although I have to admit that if I didn't have them to compare with I'd be perfectly happy with the Kobo.
I have a Kobo Aura and a couple of Kobo Mini's. I'll make this comparison about the Aura since I think the light is now a requirement since I've gotten used to it. Frankly I think the Mini is the better device though. I wish I had a lighted one.
The first and most important difference is the screen. The Paperwhite screen is much more evenly lit and has much better contrast than the Kobo Aura. Not that the Kobo screen is bad. It's probably as good as reading a new paperback and better than reading the used, slightly yellowed paperbacks I read all my life till I started reading e-books. The Kobo Aura has a good screen. The Paperwhite has an excellent screen. I don't have to have them side by side to tell that. It's just too obvious.
I just got my first Kindle Voyage today and I've read a little on it and they brag about how much better the screen is but the difference is small. In fact in some ways I'm not sure I like the screen better. It seems cooler in color and the background seems almost translucent. This is only when holding them side by side to compare. While reading they both see perfect.
Comparing the feel of the devices in my hand the Kobo Aura was the easy winner. It's small and light and a nice fit. The Paperwhite seems bulky by comparison. This is an important thing to me. I like the reduced size and weight a lot.
The Voyage isn't as small and isn't as light as the Aura and yet it feels great holding it. I think I prefer it to the Aura for it's feeling in my hand, although not by much.
The Aura has a lot more features than the Kindles. I have more control over margins and fonts and font weight, etc. The Kindles have far less control options but they really don't need them. I can fine tune the Aura to my heart's content and I still can't make it as readable as the Kindles.
Another important point of comparison is the quality of the companies behind them. I had a hard time buying my Kobo because there was a glitch on their website. I spent at least an hour a day trying again and trying to get through to Kobo support. I finally did after nearly 3 weeks. Once I got through to them they were great. The guy was willing and eager to help. It turns out my Aura has a small defect but I've decided to live with it rather than go through that again.
When I have a problem I can have Amazon on the phone in 30 seconds. It's never taken longer in the 10 years I've been buying from them. And I don't have many problems.
I buy most of my books and Amazon's books are typically 1/4 to 1/3 less than Kobo books. The selection seems better but that might be an illusion since the store and it's search features and presentation are so much better. I use the Kobo for books from other sources. If I'm going to buy a book I get it from Amazon.
Also, when I first got my Kobo I found there was no way to shop for books and avoid the porn. I'm not particularly prudish and I don't mind that there's adult material there. Amazon has plenty of that, too. But it's not always showing me porn. I see some at times and that's fine. I live in a retirement home and people are always asking me to find books for them on the computer in our rec-room and I'd be afraid to do that with the Kobo store.
I've been told that's improved a bit and the few times I've been there since it seems it may have, but it's still a bit raw.
None of these are major. If I'd had a chance to see the Aura before I bought it I might not have bought it but I'm not really sure. It's design is beautiful and I might have wanted it even if I knew about the problems. I'm not sorry now that I have it. I've had a few chances to sell it if I'd drop the price enough but I use it and like it so I didn't. It's a very good ereader and well worth having. The Kindles are noticeably better.
Barry