The mothership core was created to solve the problem of multiplayer protoss vs. protoss games, said games rarely going to mid-late game due to the warp-in abilities negating the defender's advantage. Its Photon Overcharge ability is a counter to this, along with countering zerg rush techniques. It is also intended that its Recall abilities will allow protoss players to be more aggressive.
Prior to beta patch 4 the mothership core could fire its weapon on the move; after patch 4, the mothership core could attach itself to a nexus, and only then was it able to fire a weapon. In patch 5 this ability became a simple nexus buff.
The Energize ability was removed during the beta.
The energy cost of mass recall for both motherships and mothership cores was decreased in Legacy of the Void. The developers noted that protoss struggled in base defense, so the cost was reduced, making it easier to defend multiple bases, and to play more aggressively in the early/mid stages of the game.
The core removed from multiplayer in Patch 4.0, its role transferred to the nexus itself. This was done in a bid to simplify protoss base defenses by concentrating defensive power in the nexus itself rather than a specialized unit that moves between them.