First of all a sump, even in it's simplest form, adds volume somewhere. If the main tank is 100 gallons and also you put in a 50 gallon sump, well the total number of the whole system increases to 150 gallons. With this added volume comes added stability. A more substantial volume of water takes longer to improve in temperature, salinity, or whatever parameter you want to use. So when I've said over and over, stability is essential to a healthy aquarium.
After adding volume, the following most frequent reason to add a sump in your aquarium setup would be to offer you a spot to place all the equipment that runs the one thing. Filters, heaters, skimmers- it can all use the sump. This implies less clutter in the tank or hanging from the back than it. Much more so that it may be the only option when the back with the tank fills up and you still have equipment that should be set up. Furthermore, since the sump is probably based in the enclosed stand the noise all that equipment generates will disappear also.
All sumps are fed by some kind of overflow mechanism either hanging about the back of or built into the tank. This mechanism is made so as to allow the water from your tank spill over involved with it when it gets excessive and flow as a result of the sump. The benefit of this really is how the top of the water in the tank is consistently skimmed clean. Tanks lacking any overflow often have a greasy film of proteins and oils floating at first glance with the water which can be problematic as it can block gas exchange. By having an overflow, this layer is pulled to the sump and churned into the water for the protein skimmer to deal with. Additionally, that churning likewise helps increase gas exchange - helping the dissolved oxygen amount of the water.
A sump does mean a far more stable level however tank. Marine aquariums in particular lose a lot of water to evaporation. On setups with no sump the lake level inside the tank drops as water evaporates, possibly exposing intakes or other equipment in the tank (or even corals which have grown very tall) towards the air. As well as even though everything is low enough to not be affected you'll still wind up seeing the lower level on the surface frequently which, while not exactly a tragedy, isn't pretty either.
Possibly the best advantage of a sump that isn't immediately recognizable is it offers you a good destination to introduce additives for the tank. Reef tanks typically need daily doses of calcium, alkalinity, and/or other supplements to keep the water's parameters in check. Many of these chemicals are highly concentrated and if added straight to the tank have to be added very slowly. Using a sump where one can just dump them directly into be diluted down before they go into the tank makes adding them a lot less of your headache. Likewise topping off evaporation is a lot easier using a sump for a similar reason. Relatedly, a sump produces a good spot for your heater and/or chiller considering that the localized hot/cold spots they produce will probably be safely out of the inhabitants with the tank.
Finally a sump allows for one to quicker take advantage of two techniques to enhance your aquarium. The first is a trickle filter. Basically, as the water enters the sump it's allowed to spread out and trickle over a filter media used to cultivate nitrifying bacteria. Since the media isn't fully submerged the bacteria growing on and in it receives much more oxygen and is also therefore capable of greater.
The next setup a sump shortens is really a refugium. A refugium is basically, because the name implies, a little secondary tank that serves as a refuge for algae and various microorganisms away from the hungry mouths within the main tank. The conditions in a refugium are ideally ideal for algae, which ensures you keep it growing there instead of the main tank, as well as many planktonic creatures which fish and corals like to eat. And as the population of these critters increases more and more of them will start spilling over to the primary tank to supply a supplemental source of food. But as the main population remains in the refugium the fish can't completely destroy it.