WIll you notice a difference? Well, sure - like a 2012/2013 car and a 2000 car. Everything is 'better'. I don't know the Classic per se, but the Moots will have a stiffer platform (bigger diameter tube set, 1 1/8" head tube, much beefier dropouts). Bigger steerer on the fork, etc., etc. I would think a much snappier feeling ride. Also, I assume you'll be getting new groupset, wheels - light years beyond stuff of 12 years ago. New wheels as well? Again, same thing.
Welded bottle bosses, welded seat binder, press fit and welded seat tube insert - don't add a thing to performance, but a much better alternative to what is on the Litespeed.
It'll be like going from an old school 'box' tv to a flat screen high def model I would think. They're both t.v.'s, but but there is no question which you'd rather watch.