Well tomorrow will be a week of owning my first Kobo, the Clara BW, after owning Kindles for many, many years, the last being the Kindle Paperwhite Signature edition, so I thought I would posts some thoughts - these are just my personal thoughts/ opinions (not facts!) -
To start, the feel of the e-readers -
Obviously the Clara BW is a much smaller ereader with the whole machine taking up just the paperwhite screen area (well just a bit more), exact measurements being; Kobo - 10.9cm x 15.7cm, Kindle 12.3cm x 17.2cm (screen size of Kindle is 10.5cm x 14cm so not much smaller than the Kobo overall size, Kobo screen 9cm x 12.2cm). I've found the Kobo feels much nicer in the hand for two reasons 1) Being smaller as I hold it one handed its much easier to reach the screen with your finger to turn the page 2) The bottom corners are much more rounded than the Paperwhite making it far more comfortable to hold one handed for long periods,the Paperwhites corners tended to dig into my palm after a while, the Kobo's doesn't. The kobo also feels lighter to hold, it may not be lighter, I haven't checked, but it feels it. I have to say the Kobo doesn't feel as 'premium' as the Paperwhite, it feels more 'plasticy', but that could be because the front of the kobo has a plastic edge around the screen whereas the Paperwhite is completely flat and everything appears to be behind a glass (or plastic).
Using the ereaders -
Home screen; I never use this so this will be brief, the Kobo doesn't seem to clutter the screen with books it would like you to buy compared to the Kindle, My books/ Library; Both of these are very similar however the Kobo's My Books screen does make it slightly easier to find your books in different ways, you have four tabs along the top Books, Authors, Series and collections, The Kindle has this but tucked away in a menu.
Reading - this is where the Kobo really shines compared to the Kindle, setting the fonts, font size, line spacing, margin size are all much more customizable than the Kindle, you should be able to get your book pages exactly as you like them on the Kobo, or at least much closer than you could on the Kindle. The Kobo also has a nice Activity menu under the more menu on the main screen, this tells you things like how many hours you've spent reading the current book, average minutes per reading session and average pages per minute, it also gives you a grand total of how many hours you've spent reading, I don't believe the Paperwhite gave you any of these stats. One other small thing the Kobo does which makes a big difference is it allows you to adjust the screen brightness just by sliding your finger up or down the left hand side of the screen, the Kindle you have to go into a menu to do this, although the Kindle had adaptive lighting (however in practice I found this adaptive lighting could sometimes get confused and kept adjusting itself which played havoc with my eyes at times) another thing to note is the Paperwhites had a problem with the LEDs that lit up the screen, some would display a distinctive yellow colour around the edges or corners, the first Paperwhite I received had this problem and Amazon had to replace it.
Battery life - Both ereaders give you plenty of battery life, you won't be forever plugging either of them in, but the Kobo does seem to give you a bit more, I think this could be down mostly to the Kindles adaptive lighting, it would brighten the screen far more often than I manually adjust the Kobos - although the adaptive lighting can be turned off.
Overall experience - although I've had various Kindles for years and the Kobo for only a week I can already say the reading experience for me is much better on the Kobo, not only because using a Kobo it feels more like your ereader, whereas Kindles always felt like they were Amazons and they were allowing you to use them (personal opinion) but the Kobo also feels like it was built for readers, it's the little things, the stats, the adjusting the lighting without having to stop reading etc I personally couldn't see going back to Kindles after using a Kobo now.
Oh one othe thing - it appears you can repair Kobo's yourself where as you don't appear to be able to repair Kindles - whether that works out cheaper than just buying a new Kobo is another thing!
Hope some find this post helpful and not too long and boring