The Tascam rep has posted a couple of things about the 680mkii. One said: "The main audio improvement in the DR-680mkII comes from the upgraded op-amp, including better EIN (lower noise)."
Another quote from the 680 thread: "Somebody asked about op-amps on the mkII.
The mkII uses a better sounding op-amp compared to the mkI. (BA2115 instead of BA4580)
Only the DR-70D has the NE5532 op-amp.
I thought I mentioned this before, but the Headphone knob is improved on the mkII as well - better taper and finer control."