Network Trainer = Online play
Console Trainer = Solo play
They should both work either way around, just depends on how the developer set things up.
If you want to add mods like the superman mod, simply download the superman mod SCO file, and rename it to modslot01, modslot02 etc for any other mods for however many slots said trainer can handle and then put them in either the script.img (For solo) or script_network.img (For online) and click File > Rebuild in OpenIV. Add that .img file into wherever that needs to go, do what you gotta do for PS3 and boom.
Mod Videos: http://www.youtube.com/DontGetAngryBro