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The main difference is the files that can be viewed. The Kindle app gives you access to the Kindle files on Amazon, which tend to be cheaper than purchasing MobiPocket ebooks on mobipocket.com or fictinowise.com. It also gives you a pretty large library to choose from.
If you have a Kindle, PC, Mac, iPhone (or iPod Touch), you can read the book across all the devices and it will synchronize your current location across all of them (stop reading at chapter 3 on one device, open to chapter 3 on the other).