A few weeks ago, I received a shiny new iPad 2 as a gift. I promptly resolved to set my beloved Kindle aside for at least two full weeks and do all my reading on the iPad so I could get a solid feel for how it works as an ebook reader. In the following, keep in mind that if I say "iPad", I really mean "iPad 2" and likewise "Kindle" means "Kindle 2".
Physically, my iPad 2 is a fair bit bigger than my Kindle 2. It's big enough that you can't comfortably hold it one handed like you would a paperback. I end up wanting to prop it up somehow instead. I got a smart cover, and use that to prop it up. This works, but doesn't quite get to the optimum angle. The Kindle is just small enough that I can wield it one handed.
Page turns go much much faster on the iPad 2. They're quick and easy. This also means that shopping Amazon on the iPad is very comfortable. Buying books via the Kindle is kind of a last resort, but it's just fine on the iPad. The faster screen also makes it a lot easier to jump around to different locations in a book.
The iPad does beautifully on technical books that have code samples and diagrams that just don't fit on a kindle screen. The Kindle 2 is just not useful for programming books, but the iPad is great. The larger screen is really important - there's a reason they don't sell programming books in a mass-market paperback format.
Battery life is drastically different. If I get up in the morning and start reading, the iPad 2 will run down before the end of the day. When I first got it, my Kindle 2 would go several days in a row (even with wifi on) without running down. The battery on my Kindle is dying now, so it only lasts about 1 day with wifi on, but it's still longer than the iPad 2 with its brand new battery.
Finally, the iPad 2 tires my eyes out faster than reading on the Kindle. I'm not sure if it's the lower DPI screen or the backlit vs reflective nature of the screen, but I feel a little eye strain after reading for long periods. This never happens with the Kindle.
One thing that's really surprised me is that I haven't run into a situation where I couldn't read the iPad's display. This includes lying on a hammock in full sun on a hot summer day. The trick is that you angle the screen so that it's reflecting the darkest thing possible - a tree, a silhouette, etc. It's still fairly uncomfortable, though. You squint and strain a little bit. It's a lot better than my macbook pro with anti-glare screen, because I CAN read in the bright light. The kindle, on the other hand, takes bright light in stride and doesn't have any trouble.
At the other end of the spectrum, the backlit screen means that reading in a dark room or outdoors after dark is a breeze. No need for a book light. Both the Kindle app and iBooks allow you to adjust the brightness of the screen from right there in the app, which is very convenient. I find myself doing this continuously as ambient lighting conditions change.
Calibre integration on the iPad just isn't as good as it is with the Kindle. I use Calibre to manage my ebooks because the built-in tools suck (more on this in the future). Managing what books are on my Kindle is a simple process of plugging in the Kindle and starting Calibre. Doing the same with the iPad requires an intermediate step where you load the books into iTunes and then have iTunes sync with the iPad.
Reading on the Kindle is much more focused. I find myself flipping over to check my email, twitter and RSS feeds when I use the iPad. That's never an issue on Kindle. On the other hand, I haven't played any video games or watched Netflix on my Kindle. Surfing the web on the Kindle is an emergency-only option.
All in all, the Kindle is a better tool for reading fiction and other books that don't have a lot of graphics or code samples. The iPad is better for books with graphical needs. Like the iPhone, which is at best a mediocre phone, the iPad shines as a multi-function device while the Kindle is a better reading device.