How you deal with the file system settings really depends on what type of content you visualize your site using. If you know that most files should be available for anyone to download, then leave the Default download method as Public. By the same token, if you want some control over who can access the files then use the Private method.
Public files can be accessed directly through a browser without having to go through your Drupal website. So, if someone liked a video you posted, they could reference it from their own website, allowing their visitors to see the video using your bandwidth to serve it obviously not ideal.
You need to keep an eye on this and find out if your host service provides some sort of hotlinking protection ...