I've used the Sony book reader, a precursor to the Kindle, for 2-3 years now and I really like it. Its very convenient and eliminates having to carry lots of books around. For example, I like to read on vacation and generally take with me one book per day plus 2 books for travel days. However, on our 2-week cruise last year I only took the reader with 80+ books on it!
By the way, contrary to what was indicated earlier, the display for these readers is NOT 1960's technology. They are based on the relatively new Electronic Paper Display, or EDP, which "possess a paper-like high contrast appearance, ultra-low power consumption, and a thin, light form. It gives the viewer the experience of reading from paper, while having the power of updatable information." Their display's are almost indistinguishable from printed paper (aside from color), and only use power when pages are 'turned'. (The Sony Reader claims "6000 page turns per battery charge", which seems about right after extensive usage.)
I have both the original PRS-500 and the second generation PRS-505. Even though the Kindle had been released when I was looking to upgrade from the 500, I stayed with the Sony Reader for basically two reasons: 1) The kindle was bulkier, heavier, and had a more narrow greyscale range, and 2) The books from the Sony eBook site were (and still are) about 25% less than from the Amazon Kindle site.
Ignoring the display technology, the basic difference between the Amazon Kindle and Sony Reader is the purchasing and distribution mechanisms.
You see, the primary problem that had to be solved for a book reader to be a viable product is how provide and distribute content in a licensed manner. Sony decided to use the Apple iTunes model, creating an 'eBook' store and designed a PC storefront for it. The user would then use the PC to select, purchase, download to the PC, and upload (via USB) to a connected reader. Substitute iTunes for eBook and iPod for Sony Reader and you get the idea.
Amazon's approach was different and, IMO, much more inventive. They licensed usage on one of the national 3G cell networks, and then built the Kindle with a keyboard (the Sony reader doesn't have one) and 3G network access. They then created their storefront with a Kendle-based interface and the 3G network as the connection medium. This allowed them to eliminate the requirement for a PC and allowed direct downloading of both purchased (books), periodic (magazines, newspapers), and even personal (email) content. Pretty impressive, actually.
With the launch of the Kindle V2, they've addressed all of my complaints about the physical look and feel, as well as having the improved greyscale. Even with the relative higher cost of the books, I have to say I'd currently recommend the Kindle over the Sony reader. I'll probably even pick one up.