They are two different means of nutrient control. You can run one, or the other, both, or none
The basic differences is in the approach
A skimmer does not remove phosphates or nitrates from the water. A skimmer helps prevent them from forming. A scrubber directly removes these items from the water and reduces their levels
The easiest way to think about removal from skimmers is to think about food.
Food is "organic"
Nutrients, are "inorganic"
Scrubbers and refugiums remove inorganics from the water: ammonia, nitrite, nitrate, phosphate, CO2, copper, etc, but they do not remove food particles.
Scrubbers and refugiums add certain organics to the water: Glucose, amino acids, Oxygen and copepods too.
Protein skimmers, however, do the opposite. That's why they have the word "protein" in the name.
Skimmers remove food particles and sometimes copepods from the water, but do not remove ammonia, nitrite, nitrate, phosphate, CO2, copper, etc.
Skimmers also add oxygen to the water.
So the skimmer approach is to remove fish waste and excess food before they have a chance to turn into the inorganics mentioned above. These organics, however are food for corals. Scrubbers allow the food to be present for corals and remove the excess.