In layman's terms, at least on a Gulfstream G450/G550, the cross-feed system allows the engines to be fed from either the left tank, the right tank, or both tanks (one to each engine.) the cross-feed pipes feed into the fuel hoppers, not directly into the other tank. While you can use the cross-feed system to fix a fuel imbalance, doing so would be via using the fuel (in the engines) not actually transferring it from tank to tank.
The intertank system is a valve and pipe system between the tanks themselves, allowing a fuel imbalance to be fixed by initiating/allowing the transfer of fuel from tank to tank directly.
My sources were:
G450/G550 Fuel System
G450/G550 Fuel Unbalanced