It's a subtle sound distinction between the two amps - I think both are chip-based. Unfortunately, Norm isn't really making the Classic any more, but he must have some of the parts and circuit boards lying around... with the improvements he's made in the Go-Vibe series, one can only imagine what he could do with a home-based amp design
I doubt the Classic would completely change the character of your K701s, though... I did try GlendaleViper's set on my Classic and portable source, and couldn't get it to rock out on the guitars and drums like the HP2s (on Matthew Sweet's Girlfriend). It was actually kind of surprising - I turned the volume up, but it just made things louder, not more distinct. I really did want to like the K701s, but I've been listening to Grados for more than 3 years now. Before that, I spent 9 years listening to my Sennheisers.
Best of luck on your new setup - don't let my listening preferences get you down. I think that the K701s have a lot of potential, but they haven't met their match yet. A really fast, slightly forward source or amp might complement the AKG's strengths. They don't seem to be the kind of cans that like to get down and dirty, however. Wes Phillips of Stereophile seems to be a big fan of the K701s.