There is very little distinction that a regular person makes between a dog behaviorist and a dog trainer if they don't understand the profession. You could think of a behaviorist as a psychologist for dogs, since they are responsible for fixing behavioral and emotional problems in dogs -- whereas a dog trainer would mostly use conventional dog training techniques, but may not be able to correct problem behavior.
A dog behaviorist will be someone who is either scientific, or has an immense respect and love for dogs akin to the "Dog Whisperer" many of us have seen on TV. While a lot of people are somewhat dismissive of animals and think of them as a less intelligent inferior creature: The dog behaviorist understands that a canine is a complex and emotional animal that can be affected by the world around them.
Another difference between a trainer and a behaviorist is that the behaviorist will often times come to your home to counsel you and your problematic animal -- where the dog trainer would have you come to them. A behaviorists job is to make your animal feel as comfortable as possible, while observing them and asking you probing questions to get to the bottom of your dogs issue(s).
A behaviorist may have education in: dog psychology, dog biology, dog science, etc. They might also have no other training than that which they've observed through years of personal research. Each behaviorist will have a different method of correcting behavior, much like human behaviorists will have differing methods they strongly believe in.
Anyone who is interested in becoming a dog behaviorist is in luck, due to the fact that there are so many people who own dogs now. Look into programs online or in your area to see if classes are accessible to you and if you love working with animals you may soon become one yourself.