Bushings serve two purposes. First they are intended to be references for the required diameter of the turning. That is, in theory, if you turn down to the diameter of the bushing, then the turning should match the diameter of the metal components of the pen that mate with that turning. Obviously, you need to think several steps ahead to recognize that sanding further reduces the diameter of the turning, while applying finish increases the diameter. Ideally, there should be no perceptible difference between the diameter of the turning and the diameter of any adjacent metal components in the assembled pen. Also, over time, the process of sanding and applying finish will also modify the diameter of bushings. For that reason, it's a good idea to also do some careful measurements to assure that you are getting the desired diameters on your turnings. Likewise, careful pen makers who use bushings and mandrels also consider bushings to be expendable items that need to be replaced periodically.
As you say, the second purpose is to serve as a spacer. Basically spacers simply assure that there is physical space between the wood or plastic turning and the metal ends of the mandrel so that there is room for tooling without risk of the tool hitting the metal end of the mandrel.