I prefer chicken eggs to fish eggs, personally.
The way I play them, bridge pickup only, they sound extremely similar (but then, the 4001 is a restoration with 4003 electronics). The feel is similar, slightly different necks. The 4001 is a lot heavier.
The neck pickup there's a bit more difference, as the 4003S still has the high-gain pickup, while the 4001 I have a toaster in (I'm going to have a toaster put in the 4003S). The 4001's neck pickup is a bit more "tame" I suppose. Not as mid forward or high output. Does a better upright emulation.
I don't really use the stereo on the 4001 really. In a mix, I doubt anyone could tell the difference between the two, especially since I usually only play the bridge pickup.