You're missing several advantages:
No 1 is that if the PC develops a fault, say PSU goes bang or memory fails... easy swap with replacement parts, so I can be back online within 15 minutes. you try doing that with a failed synology.....
No 2 free NAS software such a FreeNas supports far more than an off the shelf solution. try the following features, Upnp, CIFS, NFS, Itunes, RSync, Unison, iSCSI, Support for UPS'es, Samba, Dynamic DNS, BitTorrent, AFP and SNMP, amongst other features. it also supports the use of PuTTY for those who like to Telnet in.
No 3 is i can choose to put in a specific hardware raid card such as adaptec or more than one, dependent on the number of PCI slots, and configure the disc arrays exactly how i want, not have synology tell what I can and can't do.
No 4, the home brew NAS is upgradeable, more memory? faster processor? higher spec board? no problem.
No 5 One of my freenas boxes has a 64 bit PCI server motherboard, allowing the use of 64 bit PCI raid cards, I've bench marked its data throughput and its in well in excess of 200MBit/s (provided you have a gigabit card of course) which is more than what the Synology can do.