Ted got it backwards, the R-44 has a phantom powering switch for each channel (1,2,3,4) the DR-680 has phantom switchable in channel pairs (1&2, 3&4, 5&6).
Both are good machines. Which is best and what features are most important depends on your application.
I currently own both- my R-44 is and Oade Concert Mod, the Dr-680 is stock.
The DR680 has additional channels (6 analog + 2 digital) along with the features others have mentioned above (gangable channels, digital gain controls, monitor mixer). The strength of the R-44 is it's slightly smaller size, operational simplicity, and effects that can be applied for playback (or recording, but I don't know of anyone that would use it that way). The only 'effect' I ever use are a several graphic EQ's that can be applied for playback. I doubt many people here use the effects but they include a gate, an exciter, reverb, and the graphic EQs. The DR680 has none of these [edit:a reverb effect only and no EQ <<oops, this is the DR2d not DR680]. The R-44 has less physical switches on the top face that can be accidentally engaged or disengaged and those switches are less prone to being accidentally pushed. It also has a simpler interface and menu system. The metering on both is mediocre, yet perfectly functional. The R-44 is less fussy about SDHC card formating and structure, it will simply write files to any card you put into it (like the one you just took out of your camera). The DR-680 needs to at least write system files and folder structures to a new card before recording and often asks to format a card (which erases what is already on it). I have had the R-44 loose power while recording and the recorded files are visible on the card but may just need a header re-write. Not sure if that is the case with the DR-680 or not, but I have other Tascams that require file recovery software when that occurs. IMO both need an external battery (9V 'DVD' LIion works for either, but with different cables). Both can make great recordings.
I bought a DR680 primarily for the digital gain control and channel ganging feature since I use it with a Tetramic, but also for the additional channels. For me it was worth it for those features alone. I like the monitor panning feature but have not used it, for Ozpeter that's the clincher.
If you want more than 4 channels and the additional features are valuable to you, consider the DR-680.
If you only need 4 channels, want something smaller, simpler, that just works every time, aren't using an ambisionic mic and don't need monitor panning, consider the R-44.
[edit- both have limiter funtions and mid/side processing which is selectable for either record or playback, but I've not used those functions on either machine]