Thanks for your vote of confidence, Chris... This type of event happens on occasion... not often, but does happen.
The flight originated in Chicago, which means that on its arrival in the Branson, MO area (southwest Missouri, about 60 miles from Springfield, the closest larger city) it would have been roughly at a 90 degree angle to both airports... i.e., Branson Municipal and Clark Downtown. It's entirely probable that neither the pilot nor the co-pilot (Captain and Second-in-command for a more technically correct description) had been to the Branson area for quite a while, if ever. For those pilots that haven't been to a particular airport, they can brief themselves with pictorials and, of course, the ever present instrument approach charts for the particular airport.
It appears that both had thir heads "up and locked" and responded to the first visual contact with an airport, which just happened to be the Clark Downtown one, not unlike 'Nibble's' suggestion. (Both were southwest of the small town of Branson).
This was further compounded by the fact that the Air Traffic Control Tower at Branson closed at 2100 (local time) and the flight arrived about 2200, so they had no communication with the ground based facility, nor would they have expected any. The other smaller airport had no control tower at any time.
The flight would have been talking to Springfield, MO Approach Control, who assisted getting them to the area (along with GPS or other guidance) but once they told Springfield they had the airport in sight, they were on their own. The runways at both airports are nearly identical running northwest to southeast (oriented with the prevailing winds) and only about 6 miles apart. Weather doesn't appear to have been a factor.
I'll have to say that regardless of the mistake, they displayed typical Southwest Airlines pilot airmanship in landing their airplane at such a small airport safely. Although the "pucker factor" (pilot lingo for self-induced stress)) was probably fairly noticeable to both. Southwest is generally known, here in the U.S., for the quality of airplane handling among its crews. That along with the fact that the airplane was one of Mr. Boeing's finest... the B-737, first certificated in about 1965... it's well proven for structural quality.
By the way, Chris, I see your congratulatory message re: SandyRoe's reaching the 1,000 mark for 'Best Answers'... but you my friend, aren't far behind with 982! Good going!