A fuge can have several purposes. It can house algae which aid in the water's filtration, provide food for the fish (grow your own), or house baby fish when used as a hatchery. A fuge is like a seperate tank that shares the same water as your main tank.
A sump is also a tank but is mostly used to house devices like protien skimmers, wet/dry filters, and foam mechanical filters.
I have both on my system. I add top-off water to my sump when the level starts dropping. In the mean time, my main tank level stays the same. Because I have a sump, the main tank water level never changes since the tank gets it's water from the sump.
I have mangrove plants and a micro algae scrubber in my fuge. These absorb nitrates and phosphates from the water and make it incredibly clean. The algae is growing in sheets in my fuge because I leave its light on 24 hrs/day. Then I can remove some of the algae and drop it into the main tank for the fish to eat. :happyfish