It is a very complicated visa that still confuses us the most. Here is the most simple way for me to verify the differences. 

If your parents are Korean now or were Korean before, you need to check out if you are eligible to apply for an F4 visa. According to the Korean immigration law which Recently changed (2010.05.04), you can actually continue having both of the citizenship before you became 22 years old. 

 

[F4 VISA]

If your parents have American citizenship by the time you were born, you can apply for your F4 visa by showing to the Korean consulate that your parents used to be a Korean citizen but gave up the nationality. 

Here are the documents that you need to provide to apply for an F4 visa 

- A Birth Certificate (which stated parents' name on it) 

- A Basic Certificate (기본증명서) and Family Registry certificate (가족관계증명서) 

- A citizenship certificate (or foreign passport) from mother or father 

- Visa application form 

- A passport 

- A passport sized photo 

- Visa Fee 45$ (cash or money order only)

 

[Dual Citizenship] 

If you are still younger than 22 years old, you can choose dual citizenship by signing up for a written oath for disadvantage (불이익서약서) 

In that case, you need to be registered at the Korean family registry (출생신고) and can apply your Korean passport get into Korea. It is more complicated for a man due to the military service obligation in Korea. He needs to make a decision before 18 years old and need to complete or postpone military duties. 

 

Here is the useful link which explains Korean nationality questions.

https://www.youtube.com/watch?v=cjR7jfFWIiA