Depends on the series. When we first met the Borg in TNG, it was stated that their ships were completely decentralized in construction -- there was no "primary core" for any component, because every function was distributed uniformly throughout the cube. Thus, it was impossible to neutralize any of the cube's systems without destroying most or all of the cube. However, Voyager forgot this
First Contact, too. They handwave it away by having Picard tell Data to "trust" him, but Data's objection is valid.
I always figured that the area on the cube Picard told the fleet to target was where the voices in his head were originating from, ie. the Queen's chamber aboard the escape pod sphere (and possibly where the Vinculum
thingie is located as well). Though Picard didn't know about the Queen's presence there consciously at the time, he knew that's where the voices came from, and figured shutting that down would prevent the Borg from communicating and stop them. The escape hatch for the sphere hangar and the area Picard targeted are roughly the same height from the bottom of the cube, but on different faces of the cube, so they could have been burrowing a hole to the sphere from behind and to the side of it.
That's why the Queen had to evacuate before the phasers and torpedoes dug a hole all the way to the sphere hangar. It would be the one centralized weak spot on the cube now that they introduced the concept of a Queen instead of tens of thousands of drones operating as one with no central control. Maybe the cube itself wasn't even technically destroyed by the fleet; maybe she self-destructed it, as she has done with other cubes (on Voyager), to prevent the cube's capture and to give her a chance to enact Plan-B (the time travel) in the chaos.