Used the way you describe, a relay does not provide isolation between the mains and the device being powered.
It does provide isolation between whatever is providing the control signal (to open and close the relay) and the power circuit (assuming there are no other connections between them).
You can isolate the powered device from the mains using a transformer, if that is part of your system requirements.
What would be the considerations given to determine if the 120 VAC connected to the relay output need to be isolated from the primary input power?
If the device being powered does not have any specific safety features, it would be a good idea to isolate its power from ground. The reason for isolation is that the neutral wire of mains is tied at some point to earth ground. If the user contacts the hot wire (or some part that is connected to hot by a fault) a hazardous return path could be made through the user's body. Safety isolation breaks this path.
However the load device might already be designed with its own safety features (such as isolation or double-walled shielding) to allow it to be powered from mains, and in that case you wouldn't necessarily have to provide isolation in your switching device. You would want to make sure the cable between the switch and the load is routed with due regard to safety (for example, through earthed conduit).
Also remember that give the supply voltage is 120 V, isolation alone is not enough to ensure safety.