Besides obviously more efficient use of the network bandwidth, the streaming media server approach has these advantages:

Better audio and video quality to the user.
Advanced features like detailed reporting and multi-stream multimedia content.
Supports large number of users.
Multiple delivery options.
Content copyright protection.

Of course it depends on your requirements (mainly, how much data you need to push through the pipe). If you need something really fast, you could set up a server with very high data rate hard drives, for example, compared to a typical web server.