Operation of a condensate pump is like a sump pump. It works off of a float mechanism. When the condensate tanks gets enough water in it to lift the float it makes a switch. This turns the pump on and pumps water back into the boiler. The problem is the boiler may not need water at that time and it overfills the boiler. This will create wet steam. Wet steam moves slower and turns back into condensate long before it should. You lose the heating capability of steam and must run the boiler to make more steam.
The boiler feed unit is larger and holds more water than a condensate pump and does not work off of a float. The boiler feed tank will get a signal from a boiler pump controller to start and stop the pump. The boiler only gets water when it is needed. The boiler does not overfill. If the near boiler piping is correct we get drier steam. The other difference is the make-up water for the boiler is fed into the tank not the boiler. The feed water is controlled by a float. The benefit of this is as the boiler feed tank is hot due to returning condensate it helps remove the damaging oxygen before getting to the boiler. It also give a place for dissolved solids to drop out before getting to the boiler.