Board certified animal behaviorists spend years getting hands on experience before they can earn their title. This experience is often with the worst of the worst as far as behavioral problems go.
Yes, board certified animal behaviorists are veterinarians, who complete post graduate residencies in animal behavior. After they complete the residency, they are "board-eligible"; they have to pass a test (oral and written) to become board-certified. There are 49 boarded veterinary behaviorists in the US + Canada (see here for a list: http://www.dacvb.org/) These people are the equivalent of human psychiatrists, who are MDs, complete a residency in psychiatry, and pass a test. The MD (or DVM) allows the person to legally prescribe drugs.
In terms of the training and cases that these people see, the residencies are generally at vet schools. No one is going to drag their dog to a vet school and pay the $$$ for an animal that is simply badly behaved. These are dogs and cats, who the local trainers and vets can't help; veterinary behaviorists are frequently the last stop before euthanasia.
My dog was never in danger of being euthanized. For her, it was a quality of life thing, minimizing her anxiety and not forcing me to imprison her on my little suburban lot. I took both dogs to the botanical gardens today--a year ago walking in a public place with both dogs was unimagineable. I had origionally adopted her as an agility prospect, but in the end she doesn't love agility enough for the scarey stuff in the environment not to matter. When she is in a herding environment, she doesn't care about anything but the sheep, so I would have no qualms about trialing her once she was ready from a herding training standpoint.
Anyway, many dog trainers also refer to themselves as "behaviorists". Among these folks, there is a broad range of experience/ training ranging from PhDs with practical animal behavior experience to zip. So, one has to scrutinize these folks really carefully, as anyone can call themself a "behaviorist", just as anyone can call themself a horse trainer, or dog trainer, or whatever.