What does breaking the tab on the neutral side do to the outlet as a whole?
Breaking the tab on the neutral side does exactly the same as the hot side: it separates one terminal and neutral outlet from the other terminal and outlet.

I can't imagine a scenario where only the neutral would be cut.

The neutral should be cut where the corresponding hots are cut and come with their own neutral. For example, if the two halves of the outlet are controlled by two separate switches and the power comes from the switches, then the neutrals should be separate. The neutral should not be cut where the power comes in to the box with the outlet, and the wire from the box to the switch is switching the hot and returning it over the white wire.