Having been in your position not so long ago I feel I should help
Ps - I will say at this point, that I own the Synology as a result of many hours googling!
Basically it boils down to performance and customisability (is that even a word?!)
Even the entry level Synology (DS110J) which I have, has a much more powerful CPU and RAM than the D-Link, which in real-world terms makes using the system much faster and more responsive.
If you are using the NAS to simply store movie files and push them to a media player, or TV etc, then the D-Link is probably just as good***
***if using a Media Player, the media player does all the video conversion, so the NAS is basically just pushing the file to it, rather than encoding it into the correct format - which is where the CPU and RAM performance is important.
The Synology is just sooo customisable. It has its own operating system which looks like Windows, you have automated backup plans, schedules, 3 USB ports (for things such as printer sharing, backing up onto a USB Hard Drive and importing at the click of a button off a USB pen drive etc)
Mine is only a single disk enclosure, but I have a 2TB drive inside it, which backs up daily onto a 2TB WD Elements USB drive plugged in the back of it. I have the power schedule set up that it powers down the USB drive after the backup completes, whereas the internal drive is spinning constantly.
I also have the option of doing a network backup aswell, so I could, in theory, backup the full contents of the 2TB drive, or individual folders over the LAN to another Network Device or my PC. For example, I could have another 2TB drive in my PC receiving the network backup, again on a schedule of my design.
On top of that the Synology has a fantastic BitTorrent download manager built in. If you use BitTorrent you'll love it. Set off a download, turn off your PC, go to bed, go out, go to work etc, and the NAS handles the download itself and saves it on its own internal drive (or onto a connected USB drive etc) - so when you get back to your PC and turn it on - the file is downloaded ready for use. You can also get an app to add BitTorrent downloads from your mobile phone, so you're at work.....someone recommends a film thats really good... you launch the app on your phone - find the movie, set it off downloading, then when you get home its finished. I cant tell you how many times I have use this for various things.
There is the remote access.. either over web, so you can access your full drive contents (or a specific folder) from anywhere in the world, via a remote log-in. There are the iPhone and Android apps, for doing the same.
Its an iTunes server which in my house provides 3 PCs and Laptops with around 100GB of music which is shared and accessible by each device at the same time.
It also supplies two western digital TV Live hubs in my lounge and bedroom with HD Movies over LAN.
I dont know the specific specs of the D-Link, but I am 100% happy with my Synology and as I forsee myself using it for many years, the initial outlay cost didnt see so bad if Im still using it in 5 years time, when I would have guessed that I would have upgraded the D-link by then!
HTH