What are the differences?
Ideally, you are able to incident read the scene, would that be preferable for applying the zone system?
The differences should be explained in any book on the Zone System. You use reflected meters for the bona-fide Zone System. The narrower the angle read by the meter, the smaller the part of the image you can tonally place.
This being said, if you develop an eye for luminance range, you can certainly use exposure and development alterations to give yourself great negs when using an incident meter. But it isn't the Zone System, per se.
I find the Zone System most useful for when I want to highly manipulate the tonality of an image. For most things, I prefer incident metering. When things are ideal, I use both meters for the same shot. I use the incident to get a "normal" EV, and then use the spot to measure the luminance range of the composition, and to see exactly where certain subjects will fall tonally if I use that "normal" incident exposure. The spot meter tells me
exactly how much range I have, while my eyeballs tell me
roughly how much I have, i.e. flat, normal, or contrasty at the least. With this info, I can decide if I want, and exactly how I want, to expose and develop differently from what the incident meter sez.