I think William and Samuel Williamson were brothers.
William Williamson was born July 1871. His Record of Enlistment for 1895 is identical with Samuel's except that William was discharged with the rank of corporal while Samuel remained a private.
William Williamson enlisted the same day as Samuel Williamson, on November 7, 1898 at St. Louis, Missouri. Both were born in Fayette County, Tennessee. William was born July 1871. They both had the occupation "soldier", both had brown eyes, black hair and dark complexion. Both were 5' 10 1/2" tall. They both were assigned to the 25th Infantry, G Company. William was discharged July 2, 1899 (hard to read this date) from Sam Houston, Texas. He was a private and rated "very good"