The main reason I like the Accelerated Beginner Book as opposed to the Adult book is basically an esthetic preference. I personally don't like the look of the Adult Beginner book. There's too much going on on each page. It's distracting. The music is fine, though.
Some other great beginner stuff, if you want to tackle some classical--Kabalevksy Op 39. This is a nice collection. KJOS puts out a cheap version ($3.95 US). Plus, you can purchase a separate CD, I believe so you can hear what your goal is.
Also, FJH just put out a new series of Classical collections at different levels. It's called "Succeeding with the Masters: The Festival Collection". Each edition comes with a CD that has all the pieces on it. The book and CD together are only 7.95m which I think is reasonable. Each book has somethings from each time period, and I think Level 1 would be a great start for a new adult student that wanted to start some simple classical pieces. If you find these really easy then you can do Level 2, etc...