Most likely the PassKey system has not detected the key resistance pellet while the engine is running. If that's the case, the system is now in an unprotected mode, allowing you to start the engine but also not providing antitheft security. If this is like the other 1996 PassKey systems, if you disconnect the battery or lose power for whatever reason before the problem is repaired, you will not be able to start th engine.
I'd suggest starting by checking the orange two-wire connector at the base of the steering column. If you insert the ignition key and cannot read the correct resistance at the connector at the base of the steering column, the wires at the lock cylinder and/or contacts in the lock cylinder are suspect.
This may also help:
http://72.19.213.157/files/VATSSystem.html