If you are not a control panel user you will not be able to modify the Facebook or Twitter accounts associated with your public profile. If you are a control panel user you can modify the social media accounts on an individual profile under Edit > Social media accounts. 

The main difference between the two profiles is that they're part of different nations, so they're technically different profiles. You can have a public profile for each nation you join. If you want to adjust the social media accounts on your public profile but are not a control panel user, try contacting the nation to make the change.