Expanded from an earlier answer of mine

German division did not create a wholesale division of the language. Although there certainly was a political divide between the two Germanies, the cultural divide was somewhat less salient. Both the FRG and the GDR tried to lay claim to Germany's cultural heritage and there were similar pedagogy when it came to instilling proper German. The GDR public could receive Western broadcasts outside of a few areas and one of the few bright spots in the planned economy was its production of television sets. The era of Ostpolitik in the 1970s also greatly facilitated West-East contact as the GDR opened up its borders to FRG travelers (and their hard currencies). East-West travel was much rarer and more supervised in this period, but not unheard of either. So despite the political division of the country, there were some areas of common ground between West and East Germans. Language was one such area.

There really was not that much of an imposed change on the German language in the GDR. Most of the differences between the two Germanies were lexical. Some of this was partly the fault of division. For example, in the GDR Kaufhalle/Konsum were used for grocery stores while in the FRG Supermarkt became more common. Brigadier remained a reference to the military rank in the FRG, while in the GDR it became a reference to the work brigades that were all too typical of the planned economy's reliance upon shock labor to fulfill production quotas. The GDR's educational and cultural authorities also looked askance at English. The state put relatively little effort into teaching English and state media often declined to countenance the penetration of the language by such a Feind. The result of this was the GDR lexicon had far fewer English loan words than their FRG counterparts. Other relatively unique GDR words were a byproduct of the state ownership of various goods and services and their state-given names became part of the everyday vocabulary. The widespread culture of Eingaben (petitioning the state) also ensured that some SED euphemisms and jargon penetrated into daily use.

But not all of the lexical differences were political in nature. The GDR's restriction of movement and its housing crisis also helped to entrench linguistic dialects as GDR as people could not move around easily. The different nature of GDR feminism and debates over the women question meant there was less of an attempt to make German more gender-inclusive in the GDR than in the FRG. Tellingly, neither Sexist nor Sexismus made much of a penetration into official language outside of occasional attacks on the moral degeneration of the West. German wikipedia has a list of words used in the GDR lexicon.

One of the exceptions to this trend was the use of the du-form. The ruling SED party did encourage the use of the informal du (you) over the more formal Sie (you). In traditional German language convention, the informal tends to be reserved for personal settings while the formal is often used in workplace settings or ones other social situations where one party is superior, such as pupil/student or employee/boss (note this is a generalization and can be observed as much in breach in some areas- DW has a good explanation of the du/Sie dilemma ). The cultural authorities in the GDR portrayed this formal/informal division as a relic of Germany's inequitable society and did not represent the new worker's state created by the SED. In this, they were building on older traditions of German labor activism and extending the imagined solidarity of the shopfloor to the whole nation. In the workshops and in SED functions, the state encouraged the use of the informal to stress that all the citizens of the state were social equals. GDR state media often emphasized the term Genosse (comrade) and it evolved into a quasi-title for SED officials in print (such as "Rede des Genossen Walter Ulbricht," and so on). The SED privileged Genossen over the term Kameraden because the latter was more frequently associated with comradeship in an organizational context, often martial, exemplified by such the songs like "Ich hatt' einen Kameraden or "Alte Kameraden".

Ironically, this social leveling by language had something of an opposite effect upon the GDR public. Some East Germans associated excessive use of the informal with the various state functions and in private conversation the threshold for breaching the formal/informal divide was higher. Surveys after 1993 about the use of the informal noted that Germans from the GDR tended to use the informal more reluctantly than their Western counterparts. This was one of the myriad legacies of the GDR after the Wende, but such observable differences tended to be much more the exception rather than the rule.