The West Brothers are a Just-So story of modern criminology. The next three paragraphs are mostly true. But keep in mind, there's a big difference between mostly true and all true...
In 1903 a man named Will West was committed to the penitentiary at Leavenworth, Kansas. The record clerk ran his measurements through their Bertillonage system, and found that he was already in the system, having been arrested for murder in 1901.
There was one small problem, however: William West was already locked away in the penitentiary, serving a life sentence. The Bertillonage measurements were the same, the mug shots were the same, the name was the same... but these were two different people.
Fortunately, the warden of the penitentiary, Major Robert W. McClaughry, had recently learned of a new system being used by Scotland Yard, using fingerprints to uniquely identify suspects. He quickly fingerprinted the two Wests, and confirmed that they were indeed two separate individuals. And this, dear reader, is why we use fingerprints as a primary means of identifying suspects in the United States of America today.
The case of the West brothers, Will and William, is entirely true. They did exist, they did look almost exactly alike, and they were arrested and sent to the same penitentiary within two years of each other. It did indeed cause some doubt as to the efficiency of the Bertillonage system.
Some bits aren't true. First, they were not brothers. To be fair, no one ever claimed that they were related; the designation is simply a rhetorical flourish. More pertinently, there is no evidence that their case was a turning point in moving from the Bertillonage system to fingerprinting. The fact most often left out of the story is that the actual act of fingerprinting the West brothers happened two years after Will West's arrest.
This is not to say that the Wests might not have been influential in getting fingerprints fast-tracked; they did appear at just the right time in history. William West was arrested in 1901; Will West in 1903; and Robert W. McClaughry, the warden, met up a representative from Scotland Yard at the St. Louis World's Fair in 1904. A training at the penitentiary took place later that year.
However, McClaughry was a pioneer in introducing new systems to the US, and was in fact the one who had introduced the Bertillonage system to the US in 1887. It was unlikely that he was going to let fingerprinting pass him by, and in fact it is more likely that the West Brothers are famous because it was a useful story for McClaughry to use as a pro-fingerprinting tale than that the Wests actually forced his hand.