Swim trunks used to be the standard for beachwear; then surf-related attire hit the scene. Board shorts and swim trunks are both designed with water in mind, but they have different construction, especially around the waist. Either might be the best choice for you, depending on your preferences.
Board shorts are the newer subset of men's swimwear that came about in response to the rise in surf culture and all of the related types of clothing. Their name reflects their practical use for surfing, even referencing the surf board. They're made from polyester or nylon, and are therefore very quick to dry and don't get too baggy from holding excess water. They're usually longer than swim trunks and hit just at the knee, a little above or a little below. They're durable despite being lightweight. Board shorts usually have a lace-up fly in addition to a Velcro fly under that. In many cases, they have at least one side pocket.
Board shorts have a sportier look and feel, and are considered to be more youthful. Most designers and clothing companies create board shorts as opposed to swim trunks, so the designs and patterns are more stylistically relevant. Board shorts are worn for many water sports, as well as beach volleyball.
The tried-and-true swim trunk is still there, even if the popularity of board shorts seems to have eclipsed it. Swim trunks have been around since the 1930s, when they first began to show up on their own and with out a top portion of a men's bathing suit. They're more likely to be shorter than board shorts, looser and made from nylon, cotton, or polyester. Depending on the type and cut of the short, they may be clingier than a board short, as well as more absorbent of water. Most significantly, board shorts usually sport an elastic waistband and may or may not have a lace-up fly.
Because of the elastic band and the targeted demographic of adult, middle-aged and older men, swim trunks might be more comfortable and practical for your unique needs. You also may not be a fan of the length of board shorts, or where they sit on the hip (a bit lower because the hips hold them up; there is no elastic).
There is no right or wrong when it comes to swim trunks and board shorts. It's really up to your preference. Maybe you might like the pattern you find on a particular pair of board shorts. Perhaps you'll prefer the comfort of a classic pair of swim trunks.