The QuickConnect functionality is kind of a VPN which bridges the LAN and internet connection.
The DDNS setup is different. Think of it like the address of your house. The internets now can find your house (public address of your NAS), but cannot enter because there are no doors or windows (forwarded ports).
You need to forward some ports on your router to allow traffic from the internet reach your NAS on your LAN. For example: you can forward WAN port 5001 to the NAS port 5001. Now you can use https://myddnshostname.synology.me:5001 which will show the NAS' DSM webinterface.
WAN:5000 -> NAS:5000 DSM via HTTP
WAN:5001 -> NAS:5001 DSM via HTTPS
WAN:80 -> NAS:80 Website on WebStation via HTTP
WAN:443 -> NAS:444 Website on WebStation via HTTPS (certificate needed)
I am not too fond about opening ports 5000 and 5001 from the internets, because these are wellknown Synology ports that are scanned for by hackers). In my home setup, I have a static IP address and domainname https://mynas.example.com (port 443) connects to my LAN NAS DSM webinterface on port 5001 (DSM allows to have setup a (sub)domain name for the DSM webinterface).