Great answers Giulty, but after using both, I have to disagree a little with a couple points here.
You can do the Kal Penn thing have a name as short as two syllables, and people still manage to put typos into it. Honestly, I don't get it.
I had "Guilty" but no "Giulty" yet, so congratulations for coming up with something original. One down, 5! - (~15) to go, if it starts with a "G" and ends with a "y", I'm happy.
For a large media collection, the synology is a PITA to set up, their iOS apps are decent. But anytime Apple changes something, typically you wait 3-4 months for synology to catch up.
Which however doesn't break the iOS apps, nor the web interfaces, nor Plex.
A mac mini is much more powerful computer than a synology NAS. If you run MacOS server you have more capability that a simple synology NAS processor and OS, with higher performance for things like transcoding. You may want to browse the AppleTV and Home theater forum here (
https://forums.macrumors.com/forums/100/) for tips of how to handle mkv files. I was hearing that the new AppleTV plays them directly... but I dunno. Not sure how anal you are about video quality. The MacMini forum also would have some useful tips.
If you have a reasonably fast network, no transcoding will take place on the local network. And that goes for both a Mac and a NAS, whether ARM or Atom-based.
Also, I don't really know what constitutes a PITA to you, but it's a lot easier to figure out how Download/Video/Music/Photo Station works than trying to do that on a Mac.
Synology is a PITA to set up for the newbie, dunno what the comment "synology is built for all these purposes without doing it yourself" comes from. It is designed for those that need a server but don't want to spend the money on one, not there is anything wrong with that. You do have to configure the synology box, like you have to configure any computer or server. OSX server ($30 in software that runs on any Mac computer) can take some effort to set up, but is less effort that synology.. and, for the features you are looking at, I think the computer and iOS is much easier.
He asked for things that probably 10 people managed to do on a Mac, but every Synology or QNAP NAS does by default.
Hazel kinda does it, for $28.
Running a server is a much better performing option to cover all the things you want to do than a synology, and those that you may want to provide in the future, but it can be a little more money.
That might be the case, but OS X is not a multimedia storage hub and to replicate all the functionality of a NAS with third-party software on a Mac, you're beyond PITA.
I don'y have all the answers for you, but browsing the mini and home theater forums here may provide decent comparison information.
If nothing else, a Mac Mini also serves as both a Plex server and client, and you can use it as a proper computer, too.
But a Mac doesn't have something like RAID5 or BeyondRAID to protect your data to a certain degree against hard drive failures unless you add something whose cost is close to the complete NAS to it (a $189
MediaSonic ProRaid would be the least expensive option).
If OS X Server works for you, cool. If a Synology or LaCie NAS works for you, also cool.