I am not a big fan of RIE. I don't find it terribly compatible with Waldorf, which is what I am more into. From what I understand, Waldorf advocates the parent as an authority figure, whose job is to set boundaries and make decisions. I feel that some of what Janet Lansbury suggests makes the child *too* autonomous, *too* sacred.
However, you asked about AP, not Waldorf! I think that RIE tends to put too much emphasis on a too-early independence. On the other hand, I do think it is good to let babies "alone" to explore and develop, in the sense that we shouldn't jump in every five minutes with some flashy distracting toy to entertain them. (But that's hardly my image of AP either).
I think that parenting is more of a balancing act ... (not that I should be talking, with my whole seven months experience over here!!). Sure, it's good to let your child do things on his own, as RIE advocates. On the other hand, I don't believe my child is going to be stunted because Daddy likes to put him on the slide (even though he's not really big enough by himself) or help him stand up (even before he was ready). RIE is very one-sided. Her article on letting babies eat at a baby-sized table on the floor bothered me a lot -- sure, it's kind of the ultimate Baby led weaning, but my purpose for baby in a high chair is also because he's part of the family, and, as such, he belongs at the table with us. (In fact, he sat with us at the table before he was ready for food, just so he could socialize.) And it seems like a lot of her ideas are over-thought.
I also think there's a limit to Attachment Parenting (or rather, maybe I think we put too much emphasis on labeling everything). I think of myself as an AP parent, but I let my baby cry when I use the bathroom if I have to, because ... what else am I going to do? Or sometime he cries while I'm making my husbands lunch as he's heading out the door because he needs his lunch for the day. But I think that an ordinary day, with head-bumps, and small frustrations such as being in the car seat, allows for plenty of self expression,without my going out of my way to let him cry it out!
I actually just read a line in "You Are Your Child's First Teacher" about this. She said that both are balancing acts, but that when RIE parents fall off the tightrope, they are more likely to be too rigid and strict, and when AP parents fall off the rope, they are move likely to be too permissive or indulgent. Interesting, because there isn't really a lot available on RIE, at least not in this country.