In case you haven't noticed, Amazon's changed gears a little bit when it comes to its Kindle platform. If you look at Amazon's homepage, you'll notice a different message. Now that the iPad is here, no longer is Amazon just advertising the Kindle e-reader front and center, it's all about the Kindle app, which is available for a variety of devices.
As I've argued all along, Amazon is ultimately(e-books) than hardware (the Kindle), so the whole multi-prong app effort is key to its strategy of dominating the e-book market. A lot of people talk about how closed the Amazon system is (it uses its own format for its content while the rest of the industry, including Apple, has standardized on the ePub format), but in terms of accessing your library from multiple devices, Amazon is actually the most open and flexible. (Note to Nook owners: You can move an e-book from your Nook to your iPhone, but the Kindle app's page-syncing feature, which takes you to the last page you read on either device, isn't available yet).
So while we can sit here talking about walled gardens all you want, what people really want is the freedom to move their content around. I frequently move e-books between my Kindle, iPhone, and iPod Touch, which allows me to share content with other people in my family (I can continue reading a certain e-book on my iPhone and loan out my Kindle hardware to my wife when she takes a business trip).
For the moment, Apple's iBooks app only works with the iPad. I suppose this was a way for Apple to distinguish the iPad (at launch) from devices like the iPod and iPhone, but it behooves Apple to quickly make iBooks available for its other portable devices, so they can have more flexibility with their content. The fact is as much as you might love your new iPad and cling to it the first few weeks you have it, you're not going to want to carry it around with you wherever you go, especially if you already own an iPhone (virtually all the people I know who bought an iPad also own an iPhone).
In terms of the apps themselves, iBooks and the Kindle app for iPad are very similar. The bookshelf-themed front ends look a lot alike and both feature an animated page turn option (you can actually control the page flip animation better with the Kindle app). The one advantage the iBooks app has is the ability to display two pages across the iPad horizontally (so it looks like a real book), but I'm pretty sure Amazon can come up with the same layout. As far as speed of accessing the respective stores go, you could argue that Apple has the slight edge, but accessing the Kindle Store over WiFi has always worked well (right now there's no way to test iBooks over 3G because Apple has yet to release the 3G iPad).
It will be interesting to see how things play out. Apple said it had 250,000 downloads of e-books over the launch weekend and it's nice that it bundled in a Winnie the Pooh e-book with graphical images to whet buyers' appetites. Amazon has some work to do as far as displaying graphics in e-books (Editor Scott Stein has a foodie e-book that he imported over to his iPad and it failed to display the graphics properly), but it does get some kudos for getting the Kindle for iPad app out on launch day. And kudos to Apple for letting Amazon and Barnes & Noble put its apps on the iPad and allowing them to compete with iBooks.
If you're looking at pricing, for the moment Amazon still has the better deals on e-books--and better selection--particularly since Random House didn't sign on with Apple and is sticking (for now) with Amazon's $9.99 pricing model. The pricing situation is very fluid and it wouldn't shock me to see Apple put more pressure on publishers to bring prices down on the iBooks platform and it won't be long before we start seeing publishers regularly running super specials on older titles that have long been in paperback. (we see it all the time in the app store, right, so why not the iBooks store?).
It's also worth noting that now that Apple is in the e-book game, we're bound to start seeing more e-book piracy, especially if e-book prices are. When it comes to piracy, there's a tipping point, and the iPad may be it, if only because it's a lot more robust than previous e-readers--and it's also going to be popular with a younger, more tech-savvy crowd than the Kindle.
All that said, iBooks will do just fine. Those who are new to the iPad and have never used an e-reader will have a greater tendency to start building their e-book libraries in iBooks (er, iTunes) and many people will be drawn to Apple's store for its periodical selections (yes, digital magazines and newspapers, as well as comic books and graphic novels, do look better in color). But if you've already bought e-books through Amazon's Kindle Store, chances are you're going to stick with the Kindle app for purchases and use iBooks for reading free ePub e-books you come across. Also, unlike the Kindle, the iPad is good for reading PDF e-books and documents.
At the end of the day, Amazon's best hope for competing with Apple is to continue to co-op Apple's iTunes strategy and make the simple-to-use Kindle platform the de-facto e-book store and management system across a wide variety of platforms. For now anyway, it's got a healthy lead. But it could stand to be a bit more open (in terms of formats supported) and its software engineers will be under a lot of pressure to innovate as fast as Apple will.