If they were on a visual approach, the approach controller would have let them go as soon as they reported the field in sight. Unfortunately, they sighted the wrong field. The controllers may help point them in the right direction, but the pilots have ultimate control over the aircraft.
None of us was there on the flight deck, so we'll never know, but consider some of the circumstances that could have lead up to the mistake.
Winds on that day favoured arrivals from the north...so Springfield Approach probably advised the pilots that runway 14 at Branson (KBBG) was in use.
All reports indicate that the pilots were making a visual approach...probably because the 737 they were flying was not certified for RNAV.
Even though they were going to perform a visual approach, they very likely received vectors to final approach, or more likely the IAF for runway 14 at KBBG.
As they received their final position and turn from Springfield Approach, the instruction probably sounded something like "....runway is 13 miles at your 12 oclock."
So the pilots looked out the window and saw a set of lights that looked an awful lot like KBBG, and replied "we have the runway in sight"....followed by..."cleared visual runway 14 approach, contact tower 128.15"
Now some are gonna try to argue that "PLK is 6 NM northwest of BBG. True, but historical visibility for the region on that day is averaged at 10 sm. And the controller told the pilot to look for a runway off his nose, t probably about 10 to 15 miles out. he saw PLK at probably about 9 miles, and if you think you can tell the difference between 9 miles and 13 miles from 4000' through a little bit of haze, at night...
So the pilots have now fixated on the runway they spotted (runway 12 at PLK), and have dialed up Branson Tower, and likely got ..."runway 14 clear to land."
That's all they needed. The controllers all believe that everything is fine. There's no way anything looks out of ordinary. Approach is no longer monitoring him, because the flight now belongs to the tower, so the whole idea that some guy is steering him by radar...just get that out of your head. The tower knows what's going on with traffic, and the runway is good to go, so even though the tower controller can't see the aircraft (or maybe he/she could), the landing clearance is given.
Let's consider other facts that could easily cause the mis-identification....
Runway 14 at KBBG and runway 12 at KPLK are aligned a lot alike. Albeit the runway magnetic headings are 25 degrees different, if the pilots likely did not worry too much about their compass at that point. The runway was spotted, the pilot locked on to his landing target, and was monitoring his speed and decent.
Runway 12 at PLK and 14 at BBG both have a REIL....this could make it VERY easy to misidentify the airports.
The aerodrome beacons at both airports are located on the south side of the runways, and both rotate green/white....another visual clue that aided the misidentification.
Both runways are lighted. 12 at PLK is medium intensity. 14 at BBG is high intensity. from 9 or 10 miles out, and 4000'...the difference wouldn't even be noticeable.
Now the mistake has been made. I know the pilots are both gonna catch heck over it, but I doubt they'll get fired. Nor should they. They safely landed the plane on a runway at an uncontrolled airport...no foul other than that is not where they were supposed to land. And the pilot noticed the problem and averted an accident, so....
It was a build up of circumstances against them...probably some procedures will be examined and revamped, ad the pilots will get some refresh training, and maybe some fines from the NTSB. Hopefully they learn a valuable lesson.
My opinion though, is the whole thing is the fault of the FAA. Why are there two nearly identical runways allowed to be in operation as they are? Why are there no processes in place to ensure that visual reference mistakes are not so easy to make? I mean really...USA has so many bloody airports (why, I don't know) that they should have considered this might be a problem.

It's not the first time this has happened.