Difference between "NT AUTHORITY\NETWORK SERVICE" and "NT AUTHORITY\SYSTEM"

It's all about the privileges that a Service User Account has.

The Local System account is a predefined local account that can start a service and provide the security context for that service. It is a powerful account that has full access to the computer, including the directory service when used for services running on domain controllers.

The Local Service account is a special built-in account that has reduced privileges similar to an authenticated local user account. This limited access helps safeguard the computer if an attacker compromises individual services or processes. A service that runs as the Local Service account accesses network resources as a null session; that is, it uses anonymous credentials. The actual name of the account is NT AUTHORITY\LocalService, and it does not have a password that an administrator needs to manage.

The Network Service account is a special built-in account that has reduced privileges similar to an authenticated user account. This limited access helps safeguard the computer if an attacker compromises individual services or processes. A service that runs as the Network Service account accesses network resources using the credentials of the computer account in the same manner as a Local System service does. The actual name of the account is NT AUTHORITY\NetworkService, and it does not have a password that an administrator needs to manage.

For example, ASP.NET Service runs under NT AUTHORITY\NetworkService user that has not access to File System.

Reference

System has extensive privileges on your local machine (access registry, etc...), where Network Service has limited privileges on your local machine. However both of them can be used for network account. In your case I think you should use System account.

More information about difference you can find Difference between local service account and network service account in windows server with respect to SQL Server 2005

For more detailed difference check: NetworkService and LocalSystem