The main thing that springs to mind for me in comparing the two books is that they've got somewhat different aims. "Programming Ruby" (the "Pickaxe book") is a complete language reference (and hats off to Dave Thomas and co. for having done that the first time around and kept it up). My book is entirely devoted to exposition and explanation, and doesn't try to cover everything in the language.
The Pickaxe book, which I consider to be of the greatest importance in the history of Ruby (it was the book that really got Ruby recognized on a wide scale outside of Japan), was how I got into Ruby: I discovered the book quite by accident in a bookstore around the time it came out, in 2000, and was immediately hooked. I'm a big fan of it, though I haven't seen the third edition yet.
Ruby training coming up in September! David A. Black and Erik Kastner team up for fast-paced, four-day Ruby intro, in New Jersey, September 14-17. See http://rubyurl.com/vmzN or contact David.