Ok, this is my understanding. Disclaimer: I have a Zanardi, but I've only seen the Type S in pictures.
They're pretty much the same except for the following:
1. Zanardi has more regular steering wheel with airbag, not the momo steering wheel.
2. Zanardi has different seats (unique as far as I can tell). The regular NSX seats are more plush and leather, and the Type S seats are racing Recaro seats.
3. The Zanardi has red stiching accenting the interior, seat, etc. From pictures of the Type S interior, I don't see this. That said, the rest of the interior looks VERY similar, including the titanium colored accents and the titanium shift knob.
4. The Zanardi plaque (between the seats, kinda on the firewall). Obviously unique.
5. The Type S comes in multiple colors, whereas the Zanardi is red only.
Unless I'm forgetting something, that's pretty much it I think. It's pretty clear that Honda started with the Type S and just made some very minor cosmetic changes to it to make the Zanardi. I'm pretty sure the weight savings, lack of power steering, height, and tighter suspension are the same on the cars... I'd be pretty surprised if they were not in that it would have cost Honda more...
Which is why the license plate holder on my Zanardi says 'American Type S'
-Z18