In short: Not really. East Germany did develop a few dozen uniquely East German terms and slang words, many of them in reference to the country's distinct political system, but there was no genuine divergence of the German language.

There was always interchange between West and East Germany. West Germans were, albeit with border checks and general administrative hassle, allowed to cross the border, and East Germans regularly consumed West German radio and television programs, circumventing state censorship. The division between the two countries never reached the level of the Korean divide, as East Germany never adopted the Juche-style seclusive nationalism which has shaped North Korea since at least the end of the Cold War. That said, I cannot comment on the Korean language, so I do not know just how different NK Korean and SK Korean are from each other.

The most linguistic differences that existed between FRG and GDR was slang. There were certain terms that were (in some cases: still are) reasonably well known in East Germany but virtually unknown in West Germany. "Datsche", for example, indicates a weekend home, being taken from Russian "Дача" (in English, the term 'dacha' exists as an equivalent, though I believe it is usually only used in appropriate Russian/Slavic-specific contexts). West Germans would generally say "Wochenendhaus".

But even most of these specific terms have now fallen out of use, as they were either specifically political ("Bausoldat": literally 'construction soldier', unarmed conscript in the NVA; "rübermachen": literally 'to make across', illegally escaping the country and going to West Germany; "Tschekist": from the Russian cheka, official designation of a member of the Stasi) or referred to technology that was current during GDR times but is now hopeless outdated ("T34-Sport": 'T34 Sport' (inspired by the Soviet WW2-era T34 tank), common designation for the Saporoshez personal vehicle; "Polylux": generic word for an overhead projector, taken from the state-owned company that made most of East Germany's overhead projectors; "Indianerfahrrad": literally '[American] Indian bicycle', a type of Jawa motorcycle).

Today, German speakers would have difficulty to accurately differentiate between West and East Germans by vocabulary and speech patterns alone – apart from regional accents, of course. But these already existed before the GDR.