Mooring refers to the means by which a starship is physically connected to a space station or another ship. This could have been achieved via docking clamps or Mooring beams.
The USS Enterprise-D used mooring beams. (TNG: "11001001")
When the IKS Rotarran departed Deep Space 9 in late 2373, first officer Worf first gave the orders to clear all moorings and release docking clamps in order to clear the station's docking ring. (DS9: "Soldiers of the Empire")
Typically, the terms mooring clamp and docking clamp were used interchangeably, but the use in "Soldiers of the Empire" indicates that at least in the case of Klingon Birds-of-Prey, moorings, and docking clamps were different items.