I'm the resident hater of the DR-680 here, based on the now-three bricked decks I have owned or been involved with, including one that was used only indoors on AC power and bricked after less than a month.
In my experience:
R-44: Simpler, more reliable controls, simpler menus, rock solid
680: More complex controls (just read the what, six + pages of threads of people trying to figure out basic things), significant powering issues, finicky digital input, generally unreliable.
The 680 has four more channels for half the price, assuming you can get it to work reliably. That's the argument for it, plain and simple. In my experience, you get what you pay for.
If you just want a bunch of channels and don't really care about losing recordings here or there or having things mysteriously not work or having the deck just die, then at half the price for twice the channels the 680 is an awesome deal. Some of the folks here use them are professionals, IIRC, and I wouldn't discount their opinions by any means. I can only speak to my experience, and unless someone here works for Tascam corporate, everyone else here can only speak to theirs.
The accepted norm here, whether right or wrong, is that once there is enough of that anecdotal information about a product, you have something to go on. That presumption has led to things over time like none of the folks here using Zoom recorders, even though Zoom makes reasonably priced decks with lots of features (which lots of other tapers that don't read this site seem perfectly happy to use, BTW). To me, the DR-680 (not necessarily everything made by Tascam, though arguably, a lot of things made by Tascam) is in the same category as their products.