ANSWER: One means 60 years old, and the other means Korean-age 60 years old, which is really 59 years old.
The Chinese character 歲 (세) is the same 세 used in the Sino-Korean word 연세 (年歲), which means "age" or "years of age." When Koreans say "60세," they are referring to the Korean style of counting "years of age," which means an extra year is added because when Koreans are born, they are already considered to be 1 year old. In the United States, we are 1 year old after one year of being born. So, 60세 would mean "59 years old" in countries like the United States. When Koreans want to refer 60 years old in the Western way, they say, "만 60세." The Chinese character 滿 (만) means "full," so 만 60세 literally means "a full 60 years old."
ANSWER: It refers to Western-age 60 ("만 60세), which again means that it is referring to "a full 60 years old." I will explain why below.
Why does 환갑 refer to "a full 60 years old" instead of a Korean 60 years old? Because 환갑 (還甲) literally means "returning to (還) 갑 (甲)," and 갑 (甲) is referring to the first of ten "heavenly stems" (천간 天干) in the "sexagenary cycle" (육십갑자 六十甲子), which counts years in 60-year cycles by combining, in sequence, one of ten "heavenly stems" (천간 天干) with one of twelve "earthly branches" (지지 地支). A "heavenly stem" is the first character in a combination, an "earthly branch" is the second character. If you multiply 10 (heavenly stems) by 12 (earthly branches), you get 120, which means one could make 120 different 2-character combinations out of the 10 heavenly steams and 12 earthly branches, but since a "heavenly-stem" character can only appear as the first character in the 2-character combinations and "earthly-branch" character as the second in the combinations, we have to divide the 120 by 2, which gives us only 60 possible unique character combinations for the different years in a 60-year cycle.
The first year in a 60-year cycle is called 갑자 (甲子), which combines the "heavenly stem" 甲 (갑) with the "earthly branch" 子 (자), so instead of saying "환갑" (還甲), it would be more accurate to say "환갑자" (還甲子) since you "return to" (還) a 갑 (甲) year on year 11 (갑술 甲戌), year 21 (갑신 甲申), year 31 (갑오 甲午), year 41 (갑진 甲辰), and year 51 (갑인 甲寅) before returning to year 1 of a new 60-year cycle. However, if you "return to" (환 還) a 갑자 (甲子) combination only on year 1 of a new 60-year cycle.
NOTE: I accidently posted this before I finished writing it, so I will probably be adding more to the post later. I still want to explain more about how the sixty 2-character combinations are formed, but I will have to wait until tomorrow because it is bedtime.
By the way, I don't think my memory is good enough to sing all sixty 2-character combinations without looking at them.



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