With the NUC D54250 you get Haswell, HD 5000, 8GB of RAM, 120GB SSD, and 802.11ac for $634.96 (+shipping).
Here's the list:
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Intel NUC D54250
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Intel 525 120GB mSATA SSD
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Intel 7260HMW 802.1ac + Bluetooth 4.0 adapter
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Crucial 8GB DDR3 1600
OTOH, the Mini has a larger footprint, a slow mechanical hard drive, comes with only 4GB of RAM, and starts at $600.
Well, it really depends on what you're looking to accomplish with the box. In my case, I'd be looking to use it as media server, specifically running Windows 7 Media Center and Plex Media Server. So, let's compare some of the specs/costs…
First, I'd argue that 4GB RAM is sufficient for this purpose. So, let's save a few bucks on your NUC and downgrade it to 4GB. Unfortunately, it looks like that will only saves us $25, but savings are savings. So, now it's just $585 for your NUC vs $560 for the Mac Mini (that seems to be the going discounted price these days). But I think we can save a few more bucks on that NUC by eliminating the 802.11ac/Bluetooth card. For a media server, we'll want to use the built-in gigabit ethernet instead. So final price is now $551 for your NUC vs $560 for the Mac Mini.
Both will need Windows 7, so that's a wash (of course, if someone wanted to use that Mac Mini just with Plex Media Server and/or experiment with one of the XBMC live TV solutions, in lieu of Windows Media Center, that could be a significant cost savings, since you could run those under OS X which the Mini comes with, whereas you'd need to supply an OS to the NUC).
Where we'll run into a debate is with the SSD drive. You can rightly argue that your NUC's 120GB SSD is superior to the Mac Mini's 500GB spinning drive, in terms of speed. But for someone wanting to use this as a media server, specifically Windows Media Center, that Mac Mini's 500GB spinning drive is actually a superior option for DVR purposes. You'll get 4x the storage space, and I'd be wary of putting the kind of wear & tear that DVR usage puts out on an SSD. Unfortunately, we can't easily downgrade your NUC to use a 500GB (or larger) spinning drive, because that NUC case can't accommodate an internal 2.5" spinning drive.
Beyond that, the Mini also has the advantage of two more USB 3 ports, one FireWire 800 port, and a Thunderbolt port. Again, for my needs, I'd want to connect several external hard drives which house my movie collection, and the two extra USB ports will be needed.
Don't forget, too, that we'll want an IR port for remote control purposes. The Mini has built-in IR, though admittedly, I'm not sure how easy it is to utilize that under Windows 7 (again, if you were just going to run Plex, you could certainly utilize it under OS X). Assuming I couldn't use it under Windows, I'd have to hook up a USB IR sensor. Again, the cost will be a wash between the NUC and the Mac Mini there, but the NUC only having two USB ports again becomes a concern.
I think most would argue that the Mac Mini is clearly better looking than that NUC, but I'm less interested in arguing that. The Mini's internal power supply is another advantage, though. The NUC's smaller size is neat, but if it comes at the cost of requiring an external USB IR sensor, external powered USB hub, external spinning hard drive for DVR usage, and external power brick, maybe they went a little too far and should have accommodated those things internally and made the case a little bigger? Once you have to add all of those external things, it all starts to look quite messy. Admittedly, for my needs, I'd still need external hard drives for my movie collection, but you can actually accommodate a 2nd 2.5" drive inside the Mac Mini (not sure if it accommodates two spinning 2.5" drives), so you could potentially stick a couple of 2TB drives inside there or something and keep everything self-contained.
Update: I screwed up and missed the fact that this NUC actually has four (4) USB 3 ports *and* a built-in IR sensor.