Tried both methods. Bradley teaches you how to relax and work with the labor you get--hard or easy. It better prepares you for releasing your body to the process of labor, for what childbirth will feel like and how to manage it by not fighting your body.
It helps reduce some of that Fear-leads-to-Tension-leads-to-Pain cycle that birthing can cause so well. Lamaze works OK to start with, but was worthless once real hard labor started. Bradley Method was by far the best....but mostly because the teacher was great--one of those earth mother types who had a bunch of babies. She taught us how to rock and moan low and take lots of showers, she taught the partners how to do accupressure and guided imagery, she had us start taking certain herbs (with Dr. approval) two weeks before delivery (I want to say it was evening primrose oil capsules and calcium supplements with rasberry leaf tea, but can't remember...all stuff the Dr. and/or midwife said "couldn't hurt" even though she didn't believe in it)--every pain management technique or labor progression technique known to man this lady covered! She was also cool about the whole labor intervention thing and worked really good with all the local OBGYNS and Mid Wives.
If you are the person who thinks independently and prefers more info--Bradley is for you. It also gives you more info on diet and lactation than Lamaze classes did. However, it is a bit dogmatic... So remember that you need to take from it what will work for you--as mentioned--some of the Bradley teachers can be anti hospital/anti epidural.
A mom should not go into a birth with a lot of preconcieved expectation, other than the hope of a healthy baby. Each birth is unique, and there are places were interventions and epidurals are important and necessary. It is more important that you work with your health providers to have a healthy baby than to have the perfect birth experience. I was a support person for a very rigid friend who needed pit to progress after a long, stalled delivery, and subsequently badly needed an epidural to rest. She fought each intervention and made the birth harder/longer than it should've been. Once she agreed to the epidural and had a good rest, labor progressed, and she was able to deliver the baby.
I had all my babies vaginally, the first two were natural/no epidural. Had them with the help of a wonderful midwife in a hospital birth center--had to drive across the Bay Area to get this. My littlest baby, my middle child, was transverse and a very difficult delivery which involved inversion. Eventually the midwife was able to turn her posterior presentation. Bradley methods sure helped me deal with this one. With the last, I didn't want the long drive, so I did the local hospital labor room with an OB, which was also very nice. They did give me the IV the minute I walked into the room, which was OK--you can still move around and it's not a big deal like Bradley makes it out to be. I did ended up insisting on the epidural during transition, the Dr. should've said no as I went through all that "stay still" bent over a huge tummy epidural proceedure only to deliver 10 min after. It had been awhile since I had done the Bradley method and had forgotten that a sure sign that the baby is coming is that the mom gives up and thinks she can't labor any longer...all part of the process of giving up control to the instinctual labor process.
Each birth I've seen--including my own--has been unique. Bradley gave me the most useful material and methods to deal with the fear and pain of labor. Found the books and info from Dr. Bradley himself to be rather weird and outdated (partner perinium massage with hot oil--no thank you) but the information and perspective you learn through the Bradley method is very useful. The big difference to me was that Lamaze only works if you are going to have an easy labor. Bradley gives you tools to face all types of labor. But use it as a tool to help you birth a healthy baby, not as a rule book for the perfect birth.