Dry just adds nitrous, and in the case of the ZEX set-up, raises fuel pressure to your stock injectors to provide additional fuel for the nitrous.
Pro: generally cheaper/easier to install
Con: greater risk of running lean due to possible inconsistent distribution of nitrous among ports (i.e., all injectors may be adding equal amount of additional fuel but some of the port may be getting more nitrous, and thus need more fuel that others).
Wet set-up uses jets to spray both nitrous and additional fuel.
Pro: Generally safer to use over dry as nitour and fuel are sprayed at the same location and thus less chance of running lean on a particular cylinder.
Generally safer at higher power levels due to more even distribution.
Con: generally more expensive and slightly more time consuming to install