There's nothing magical about a sump - it's basically just a second aquarium that's plumbed into your main display tank. Your basic sump design will drain into a filter sock of some type, empty into a protein skimmer chamber, pass through some form of chemical filtration media, possibly flow through a refugium, and finally drain into the return pump chamber. The value of a sump is largely dependent on what you do with it. That said, a sump does a couple of major things for your tank:
1) It increases the water volume of your system.
More water is always better, for a couple of reasons. The biggest reason, though, is that the more water you have, the more volume there is for toxins, pollutants and contaminants to disperse within. I'm going to use some made-up numbers here - they're not intended to reflect reality in any way except to show the impact that having more water can have. Say you have a fish that, net, makes .5ml of nitrate per day. In your current tank, without a sump, that would correlate to a 3.8 ppm nitrate level. Add a 15 gallon sump, though, and without doing anything else, it's only a 2.7 ppm nitrate level. Not a huge difference but it is a difference.
2) It improves gas circulation.
Your aquarium inhabitants need oxygen to survive, and they need water that isn't saturated with carbon dioxide. Photosynthesis does take care of some of that, but the largest contributor to the diffusion of gases is the surface area of the water in your system. Adding a sump will give you a ton of extra surface area for gas exchange.
3) More flexibility in equipment.
Trying to cram all of your equipment into the display area of your tank is a bit of a challenge, and you're limited by the available real estate as to what you can do. Add a sump and you increase your options. As an added benefit, you don't have to clutter up your display with power cords, heater tubes and other things that are clearly not corals. Other equipment choices that aren't really usable without a sump include media reactors, calcium reactors, brine shrimp hatcheries, phytoplankton reactors, UV sterilizers, refugiums and more.
4) Space for ugly, useful things
There are plenty of things - like a big ball of chaetomorpha algae - that are great for your reef tank but undesirable for a display tank. Other possibilities include live rock rubble, algae turf scrubbers, Marine Pure blocks, Xenia farms, Aiptasia farms (not recommended!) and more. Basically, if it's ugly or problematic but good for your tank - into the sump!