Mac Mini with OS X Server v. a dedicated NAS
The question is a bit vague, but having generally considered this idea myself here are some equally vague thoughts...

Disk Capacity

With a NAS you're (generally) limited to N drives, where N is the number of bays. So a 4-bay NAS will never go beyond 4 physical drives (There may be some NASes that allow external drives, I don't know of any though).

Your Mac Mini solution (thanks to daisy-chaining Thunderbolt enclosures) will be expandable to as many drives as you like (pretty much).

In general, I'd suggest this isn't a huge advantage for the Mac Mini as it may seem at first, because unless you're doing something unique, you'll probably only want a few drives anyway, and will probably end up swapping them out for larger drives later on, rather than continually adding extra drives on. After all, you probably won't want really old drives in there (they'll eventually fail, and be slow, and less power efficient, etc), so replacement after a few years seems likely.

Some years ago I built a desktop-case based NAS with 4, then 5, then 6, then 7 drives... and a very convoluted setup to let me use all the space on all of them for a RAID device. I'm now back to 3 drives and wouldn't trust the old 320-750 GB drives anymore, even if I had space for a 7-disk NAS. A 2TB disk isn't that expensive, and covers 2-6 times the capacity of my old drives, so just 2x2TB drives is as much space as I had in my 7-disk NAS 4 years ago, and much easier to deal with!

Also, if you're planning to use RAID, you'll want to have similar if not identical drives, even if your RAID system supports mixed disks. Fewer things tend to go wrong with simpler setups. So plan for complete replacement of all disks after 2-4 years.

Ease of Configuration / Re-configuration

This kind of depends on how you want to set it up, but both Mac OS X and the firmware on NAS devices make it pretty easy to configure and reconfigure disks.

Most NASes have a web-based admin/setup, while OS X obviously will require you to log in with keyboard/monitor or some kind of remote desktop thingy.

Some NASes will let you use mixed disks, while most will treat all disks as if they were as big as the smallest in the set. Mac OS X, like most 2- to 4-disk NASes supports RAID0 and RAID1, and also a concatenated. Some NAS devices, especially if they're 5+ disks, will allow RAID5, RAID6, RAID1+0, etc. I've not used the MAC OS X RAID software enough to know how it handles mixed disks.

Quite a few NAS devices will let you expand a RAID array without losing any data, either by adding an extra disk or by replacing the disks in the array. Almost none support reducing the total size or number of disks in the array. I suspect this is the same for Mac OS X's RAID implementation.

Many NASes actually won't let you treat each disk individually, which Mac OS X will, so if you want to just have a collection of disks you can plug in, each independently, with different data shared on each, it might actually be advantageous to use the Mac Mini. Though with a headless machine and daisy-chaining of drives this won't be totally straight-forward.

The RAID performance of a Mac Mini will not be as high as a hardware RAID chip, though these days its not too different. In any case, for network read/write speeds the limiting factor will be the network and protocol (AFP/Samba/etc) rather than the physical disk speeds. I think its safe to assume you're not after super high performance given you're looking at a Mac Mini vs consumer-grade NAS devices.

Power

The Mac Mini itself is very low power, as are many NAS devices. They're pretty much all based on laptop-style chipsets for low power usage. I think the main difference you'll see is in the external enclosures if you go the Mac Mini route. Each thunderbolt enclosure will likely have its own power supply/converter, and will likely use more power than the equivalent disk in a purpose-built NAS. All I can really say is check the specs of what you're after, because some disk enclosures I've used in the past have awfully inefficient (really hot) power supplies, using much more power than the other more efficient ones. This can add up to quite a bit of extra heat + electrical costs in the long run.

If power consumption/cost is a concern, you might want to consider 2.5" laptop drives, which use less power, and can often be "bus-powered" rather than needing their own external power supply (though I don't know how far you'll get with Thunderbolt).

Most NASes as well as the Mac Mini support putting drives to sleep when they're not in use, so will run at a lower power level when idle.

Neatness

May or may not bother you (or, quite importantly, other people you live with!), but imagine a nice simple box with only power and network cords (ie the NAS), compared to a mac mini + 2 or more external hard drive enclosures, each with their own power supplies + cable connecting them to the Mac Mini, on top of its power + network cords... or just power if you're using wifi). And an extra power board to plug in all those power adaptors. It might not be so bad for 1-2 bus-powered drives, but will get pretty messy if you're using big desktop-sized drives with external power and lots of daisy-chaining... If you live with others, as I do, they may have strong opinions about which option is better ;)

Future Proof-ness

The only real difference in future-proof-ness that I could think of was the SATA interface for the drives. Pretty much every NAS uses SATA drives, while the Mac Mini is using a Thunderbolt (or USB3) interface, so you will probably be able to replace your enclosures if/when SATA gets replaced by the next big thing. Honestly I wouldn't worry about this too much unless you really need this one specific piece of hardware to last 5+ years.

Thunderbolt may provide improved network speeds in the future, if/when you switch to a  10Gbps LAN, if you get a 10gbps thunderbolt/lan adaptor when such things exist. Again, depends how willing you are to fork out for a whole new NAS, rather than just an adaptor, if/when that actually becomes a problem.

Extra Features

The Mac Mini is a whole computer, so you can run anything on it you could run on a normal computer. That gives it a second life if your needs change later on, or a dual-life if you hook it up to your TV or something, rather than running it headless.

Most NASes can run some limited set of software, usually multimedia applications for network streaming of photos/video/music. The Mac Mini can do all that of course, and more, but the basics are definitely there for most NAS devices (e.g. DLNA media streaming, or as the network store for other media centres).

Final Thoughts

I think the main differences between a dedicated NAS device and a Mac Mini will be

The Mac Mini is


More re-usable (because it's a computer in its own right — either now, or after you replace it with the next NAS!)


The NAS is (probably)


Cheaper (add up all the costs, including delivery, of all parts you'd need for each configuration)
Neater/tidier (all drives are internal)


Both will be


More than capable of sharing several TB of disk space at gigabit speeds
Able to run a few extra services if you want (multimedia, streaming, network utilities)
Able to be set up easily, and reconfigured for drive changes/replacements/extensions
Easy to check status, health, etc
Low power usage
Quiet