@WOODEN_PICKS 's answer is correct. I would like to elaborate on it.
In order to better understand the difference between "一年零一天" and "零一年", you need to learn to correctly "断句" (split words).
"一年零一天" = "一年/零/一天"
"零一年" = "零一/年"
The usage of "一年/零/一天" comes from how the Chinese read big numbers in Math in Chinese, for example, the number "101" is read "一百/零/一",the number "10001" is read "一万/零/一".
"零一/年" is short for "二零零一/年", ie. the year 2001. Once again, it comes from how the Chinese read big numbers in Math. In theory, "零一/年"can refer to any year ending in "01", i.e. "2001年","1901年","2101年" etc. The actual year it stands for depends on when the writing happened.
After you learn how the Chinese read big numbers in Math, You will have a better understanding of this usage.
零一年,我去「了」内蒙古的桦木沟拍摄