1) More storage. Most people's computers don't have terabytes of storage.
2) Easier access. On a NAS, any computer can grab media as needed, instead of having to transfer files from one computer to another with a flash drive or email.
3) Better reliability - a NAS is going to be less likely to die, and on top of that, are often run in a RAID configuration that means that even if a hard drive fails, the data can be saved.