The 3 and 4.2X AO zoom scopes are a little optically simpler than the model 580 6X , which in some AO literature is described as " high resolution". I would have to say, I agree with that but the resolution difference is not huge.
The Cycloptic has yet an edge in resolution over the 580, largely due I think to it having a slightly better colour corrected image plus not having the complications of the zoom optics. The Apochromatic primary objective of the Cycloptic is fairly unique
and is something one would only expect to find on much more expensive and celebrated microscopes.
The advantage in resolution that the Cycloptic has is to some degree outweighed by as Pete noted, slightly poorer contrast and the more you burden it with aux. optics to increase the magnification the contrast dips further; less so with the 580 once over 25X, the Cycloptics limit with the standard 10X eyepiece.
The Cycloptic has a fantastic array of fractional apochromat objectives though, 3/4,2/3,1/2,1/3 and 1/5X. Some of these are very hard to find but they are out there, so for special purposes, especially on a boom it speaks to the instrument's remarkable versatility. It is light, small and has a much smaller footprint than the 580, so if you wanted to outfit yourself with a very fine stereo microscope with multiple high quality accessories cheaply; the Cycloptic is the way to go.
If you want the convenience of zoom, with easily obtainable magnification up to 60X; the 580 won't disappoint, especially at the prices they can be had for. They sit tall on a bench and are a little big and heavy on a boom but so is any other zoom stereo of it's capability. Lots of accessories for it too.
One needn't be hampered by a 23mm tube and a 20mm f.o.v. either, so a word about field of view and magnification. For quite a while now, I have been using the Bausch & Lomb U.W.F. 15X eyepieces in a Cycloptic. These were made for viewing military reconnaisance photos through a B & L 240 stereozoom over a light table. They are outstanding eyepieces and turn the Cycloptic into a true widefield stereo, very close to the standards one would get with modern scopes costing many many thousands of dollars. The real field through the B & L 15X U.H.F. eyepieces is within a fraction of that captured by the 10X 20MM f.o.v. eyepieces common to many microscopes, so the apparent field of view is out around the edges of your shoulders. The field is stunning and remarkably flat and distortion free. They also work just as well on the 580 but bear in mind that taking any stereo zoom to 90X will reduce contrast quite a bit. On the Cycloptic the limit is 40X with the standard 1X main objective and 15X eyepieces and I have not tried those eyepieces with a 2X aux. lens.