It's not a matter of British vs. American English. They are the same profession now, but in the past barbers were very different.
In the past, barbers were surgeons. Anytime who needed an operation had to go to a barber. Yep, that means barbers were doctors who also cut hair! As the medical systems English speaking countries developed, medical doctors started doing surgery instead of barbers. Barbers continued to do some medical treatments, like cutting the skin under people's hair to bleed them.
More recently, barbers stopped doing bleeding medical treatments and only cut hair and shave people.
Hairdressers are a newer profession and of course they only cut hair. They don't have the history of doing surgery. So you could say that modern barbers and hairdressers are almost the same, but in the past barbers were people who did surgeries. Today, barbers only cut and shave men.
P.S. - did you know that in the UK when a doctor specializes in surgery and becomes a surgeon people stop calling him "Doctor So-and-so" and start calling him "Mr. So-and-so"? It's because English surgeons are still members of the Barber's college.