I've done both too, and it really depends - exactly as memyselfandi says.
A NAS is primarily designed to share files. Manufacturers then tend to have plugins available for common home requirements - bit torrent client, usenet, cloud backup, etc.
If you want it to do something and there is no plugin available, then it's basically tough luck.
They come at a price premium, but you basically insert disks and power it up.
A home server can be pretty much any OS you want, with any of the software available for your chosen platform. If you plan the hardware, you could put ESXi on there and use multiple OSes, depending on your tinkering requirements.
This comes at the cost of your own time to set it all up and maintain it - and the time to learn about all this stuff.
A NAS isn't inherently 'better' for backups than a server. If your files are not stored on the NAS or server, or are stored only there - you're still at the same risk.
Your stepson may be referring to RAID, this is some protection, but should never be considered a backup.
A well considered and planned home server can easily be setup with resilient RAID in place. By the same token, a NAS could be setup with RAID0 and put you at more risk than no RAID at all.