It was the Cold War and tensions were running high between both sides. It was nothing more than a show of force - both sides had to show off their toys, and then afterward they took them and went home.
I too was stationed in Berlin but in the 1970s. And the stand-off was NOT about prisoners escaping. That problem was addressed by building a higher wall and adding more security measures on the east side. Believe it or not, the wall used to be much shorter and the guard towers only had a single guard manning them.
But the problem with escapes forced them to build a high wall with a large concrete culvert along the top that prevented anyone from pulling themselves over. Other improvements included a minefield, a dog run, a walkway for mobile guards to patrol, and hedgehogs (those steel anti-tank obstacles that resemble jacks from the old childhood game of ball and jacks).
But the most important change made was the guard towers were much higher and manned with two guards. They had to have two guards because of back when it was only one guard, the guard would escape.