There were two Winston Churchills, the British Prime Minister and his (very) distant cousin American novelist Winston Churchill, a popular and well regarded author of the early 20th century. I believe these two either met or corresponded on at least one occasion.
Also, in the 20th century, separated by a few years, two men whose real names were Joseph McCarthy, popularly known as "Joe"; one, a longtime manager for the New York Yankees baseball teams; the other a U.S. Senator from Wisconsin known for his investigations into Soviet infiltration into and influence in U.S. government during the post World War II era.
There were two men who achieved a modest level of prominence in their fields with the real name of Noble Johnson in the past century. One was a U.S. Representative from Indiana; the other an Afrcan-American actor who appeared in dozens of movies from the period of the First World War till 1950.
I'm not sure that this is wholly in the spirit of the OP but I'll give it a whirl: in the movies, back in the Fifties, there were two Hollywood actresses with the name Vera Ralston. The first one, coming to films later, became Vera Miles (she was, literally, born Vera Ralston); while the other, a Czech skating champion had her last name (not sure about the first) changed from Hruba to Ralston. Interesting factoid: both actresses appeared in films opposite John Wayne...and that's all folks!