I've got a Kobo Vox. Not having had or used a Kindle I cant obviously contribute to the either/or debate but thought I'd still pop in and leave my opinions...
The local library link works brilliantly for me. The range of titles isnt huge but it's still more than big enough for what I'll ever need. It means you can 'borrow' reference books for a day or two, or travel books for holidays and while there is obviously a wait for new titles, they do keep adding new stuff and the maximum wait is 3 weeks. No problems there.
The Kobo store seems pretty comparable to Kindle store for books - the same free titles, pay titles seem to be slightly more expensive though although usually only 25p-50p more.
I make good use of the app features and its also my first venture into the "android" world. Again, something which I have really used to my advantage with some really useful apps.
I have also copied all my music onto it, and it syncs brilliantly/easily to my gmail, google calendar etc.
I don't know how many of those features are common to the Kindle too though
The only negatives I would have would be battery life - if permanently connected to Wifi, downloading books or apps, using for facebook/internet surfing/email I need to recharge daily. With Wifi off and just using for reading it lasts a week for me though. I tried it out for a flight I've got coming up and in total I read for nearly 6 hours before the battery started to give out.
The other thing is that it has 'factory reset' itself twice

I'm currently trying to establish whether mine is faulty and to be fair, their customer communication/resolution system is a good one. This causes you to lose your apps but after the first time it happened I learned how to back everything up and it was a 2-minute job to restore everything after it happened the second time. Less than ideal but as I say, for me, the benefits of the Vox definitely outweigh the negatives.