the two names aren't the same name.
DNS is domain name service and is used to map a name to IP address. For example, abc is a child domain of def. def is a child domain of ghi which is the root domain of the forest. So the DNS zone, DNS service is authoritative for encompasses abc.def.ghi.com Domain names are logical name spaces not physical.
The computer name is the name of the computer. some name them server, some name them NS if their name servers, some give it a combination of numbers and letters, blah blah. It's just a name for the physical machine.
To confuse you further, if your running active directory in a Windows 2003 environment, you actually have 4 names going. The Domain Name, the local LAN domain name, the Netbios name and the computer name.