The benefits of an under mount sink compared to a drop in sink are both aesthetic and function.
As far as function goes, under mount sinks are easier to keep clean. Because of the under mount styles don’t have the lip on the countertop as the drop in ones. When you are wiping your counters, if there is a lip it makes it more difficult to get the debris over the lip and into the bowl. As time passes, the food particles can build up under the lip of the sink. Over time it can become unsightly, smelly, and unsanitary. Under mount sinks don't have this problem because there is no lip. The food particles just fall from the counter directly into the bowl.
Without the function, aesthetics is another factor. The main reason people choose under mount versus drop in is the under mount sink can bring a minimalistic look to the kitchen. Without a lip around the edge there is nothing "sitting" on the countertop. Even though the lip of a sink isn't technically an object sitting on your counters cluttering them up, it is raised up so too the eye it just looks like another thing taking up space. You might not have ever thought about the lip of a sink this way, but when you compare the two it is obvious.
Plus, the faucet doesn't have to mount to the sink so you don't have that space in the back of the sink for the faucet to mount. You can choose exactly where your faucet is going to sit. To make the space look clean and well maintained, less is better.
Under mount sinks are easier to use, look nicer and less cluttered, and are usually better quality that a conventional drop in sink. Just make sure you are installing a solid surface countertop such as granite before you order one.