First of all a sump, even in it's simplest form, adds volume somewhere. When the main tank is 100 gallons and you put in a 50 gallon sump, well the volume of the complete system rises to 150 gallons. With that added volume comes added stability. A more substantial amount of water takes longer to change in temperature, salinity, or whatever parameter you need to use. And as I've said repeatedly, stability is essential to a healthy aquarium.
After adding volume, the subsequent most typical reason to incorporate a sump in your aquarium supplies would be to give you a destination to put all the apparatus that runs the one thing. Filters, heaters, skimmers- it can all go in the sump. This implies less clutter within the tank or hanging off the back than it. Much more so it will be the only option when the back of the tank fills up but you just have equipment that needs to be set up. Furthermore, because the sump is likely based in the enclosed stand the noise everything equipment generates will be reduced too.
All sumps are fed by some type of overflow mechanism either hanging on the back of or constructed into the tank. This mechanism is made in such a way regarding let the water from your tank spill over in it in the event it gets excessive and flow down to the sump. The advantage of this is how the surface of the water inside the tank is consistently skimmed clean. Tanks lacking any overflow usually have a greasy film of proteins and oils floating at first glance of the water that is problematic as it can block gas exchange. Having an overflow, this layer is pulled into the sump and churned into the water for the protein skimmer to take care of. Additionally, that churning likewise helps increase gas exchange - helping the dissolved oxygen degree of the lake.
A sump does mean an even more stable water level however tank. Marine aquariums particularly lose plenty of water to evaporation. On setups with no sump the lake level within the tank drops as water evaporates, possibly exposing intakes or other equipment inside the tank (or even corals who have grown very tall) to the air. As well as even if it is all totally low enough to not suffer you still wind up seeing the low level from the outside frequently which, whilst not exactly an emergency, isn't pretty either.
Possibly the best good thing about a sump that isn't immediately recognizable would it be gives you a good place to introduce additives towards the tank. Reef tanks typically need daily doses of calcium, alkalinity, and/or other supplements to keep the water's parameters under control. Many of these chemicals are highly concentrated and when added right to the tank need to be added very slowly. Creating a sump where you can just dump them directly into be diluted down before they enter the tank makes adding them significantly less of your headache. Likewise topping off evaporation is simpler using a sump for the same reason. Relatedly, a sump produces a great place for that heater and/or chiller considering that the localized hot/cold spots they produce is going to be safely out of the inhabitants of the tank.