They are basically the same thing. Server is a general term. A NAS is a file server.
I definitely recommend building your own Home server if you like building computers. You will be able to pick your own parts, install your own OS, and use it for more then just file storage.
Some people recommend using old parts. And you can definitely go that route if you want. But if your willing to put some money down you can get some really good performing and low powered components for a home server.