A system admin is responsible for the server itself. Installing, upgrading, maintaining the operating system, user maintenance, security on the server, documenting the system, monitor system activity, etc.
A network administrator is responsible for the LAN/WAN: switches; routers; hubs, etc. If a new server is installed, the network admin is the one who assigns and IP or subnets a network. They will monitor the network for performance and use sniffers to trace activity. They are also responsible for security outside of the server. Firewalls may or may not be included on this team depending on the company setup.
At my last job (Fortune 500) the system/network admins were completely separate, and are separate in my current job too. In a smaller company they will be the same because a company won’t/can’t afford two salaries.
Even if you work in a large environment though, a system administrator should have knowledge of networking. Though I have worked with other sys admins who take the opinion that networking is someone else’s job so, therefore, they don’t need to know it – and don’t. But again, it is good to know.