In 1952, the editors of the Wörterbuch der deutschen Gegenwartssprache (Dictionary of the current German language) were facing a problem: They were tasked with documenting the German language as it was presently being spoken across the country, but only seven years after the end of the war and only three after the Federal Republic of Germany to the west and the German Democratic Republic to the East had been established, they were already detecting slight variations in everyday language usage. Some words were limited to one or the other "because of the different social-political structure of the two states", as the dictionary put it, and the editors therefore added "DDR" (=GDR) or "BRD" (=FRG) to some words which were in turn divided into three groups: new loan words, new compound words and new meanings of words. So what were the sources for this divergent development?
Obviously it was the political division for one, which began early after World War II, reached its peak after the Inner German border was massively built up into a complex system of walls, military installations, minefields and various other systems designed to make any unauthorised travelling between the two states virtually impossible. But at least as important were the different political systems and the different cultural influences: Whereas West Germany obviously saw a lot of ideas, practices, imagery and vocabulary seeping in from e.g. Italy, France, the UK and the USA, for East Germans this was limited instead to the Soviet Union and its Warsaw Pact neighbours. In both cases, this cultural exchange mostly happened via personal travel, the consumption of respective media and the large amount of military presence in either country: American, British and French in the west, Soviet in the east.
There are a couple of caveats to that, however. The two newly created linguistic communities didn't develop at the same pace; instead, the wide distribution of West German media throughout the GDR (with the exception of its northeastern- and southeastern-most regions where western TV signals had a hard time to get through - the Dresden area was lovingly called Tal der Ahnungslosen ("Valley of the Clueless") because of that) had no counterpart in West Germany. Up to 70% of the East German population regularly consumed western TV and radio, but on the other hand eastern media were hardly consumed at all in the west. And another difference was the cultural exchange with the respective foreign military presence: Whereas in the west, French, British and especially American personnel were an important factor in importing cultural ideas and language from abroad, both East German and Soviet authorities were keen on keeping the communication between the GDR populace and the Soviet soldiers stationed there (370,000 in 1982) to a relative minimum.
All that being said, let's get to the actual question: How did the language diverge? I'll focus here on linguistic development in the GDR because without spoilering too much: Most of the GDR-specific vocabulary vanished pretty quickly after reunification, and today's German language is for the most part the one developed by and in the former West Germany. Grammar and syntax remained identical in the two Germanies, with vocabulary the only real difference - and even there only in parts. Many of these new words were actually ideologically inspired and were purposefully created and introduced in order to replace either words who were seen as not compatible with socialist ideology or words who were identified as being too "western" in origin. Those could be newly minted euphemisms like Bürger im höheren Lebensalter ("citizens of higher age") instead of Rentner ("pensioners") or newly created words like Schallplattenunterhalter ("LP entertainer") for a DJ or Niethose ("riveted trousers") for jeans. Then of course there were words that were either wholly new to the German language or were understood differently than in the west. One example for that is Traktorist, i.e. a tractor driver in East German collectivised farms which was a direct loan from the Russian Тракторист. Another was parteilich, which in the west was commonly understood as "biased" but which in East Germany could also be understood as "connected to the ruling socialist party“.
The official language of the GDR also tended to be strongly formulaic and heavy on the nouns, which might have been a Soviet/Russian influence as well. Bündnisse („alliances“) were always fest, unerschütterlich or unauflöslich („firm, unshakeable, permanent“), the Soviet Army was ruhmreich („glorious“), you looked vertrauensvoll („full of trust“) into the future or were part of an initiativreiche Massenbewegung („mass movement that is rich in initiatives“). Some of that seeped through and became part of everyday speech as well, but for the most part it was limited as being an official register of speech mostly used in official writing. Official GDR writing was also heavy on the nouns which is a fault of all official and/or bureaucratic German writing but was especially noticeable in the east. Check out this excerpt of a statement by the Ministry of Education and Youth from March 1990, where many of the things I’ve mentioned before come together: [Wir wollen] das Konzept zur Weiterentwicklung unserer allgemeinbildenden Schulen (Thesen zur Schulreform usw.) verstärkt propagieren, um zu verdeutlichen, daß die bisherige Enge aufgehoben und Raum für breites Schöpfertum geschaffen wird. („We want to propagate more strongly the concept to the further development of our general-education schools (theories to school reform etc.) in order to make it more clearly that the previous narrowness is going to be abolished and space is going to be created for wide creativity.“ Note that translating Schöpfertum as creativity doesn’t even begin to capture the bureaucratic nuances of that word). Especially propagieren and breites Schöpfertum are clear markers of GDR speech here.
When the GDR eventually failed and Germany got reunified, the socialist party rhetoric was naturally the first to go, as well as all the verbiage associated with an economic system that was no more. There were no more Thälmannpioniere (members of the official GDR youth organisation for kids aged 6-14), no more Intershops (a chain of government-owned retail stores which only accepted hard currencies, which excluded the East German mark) and no more Landwirtschaftliche Produktionsgenossenschaften („agricultural production cooperatives“). GDR words where a direct western equivalent existed lingered on for longer, but by now even in the former east you’re more likely to work in a Team instead of a Kollektiv or to live in a 3-Zimmer-Wohnung instead of a Dreiraumwohnung („apartment with three rooms“). Some words still survive, if mostly only with older speakers, like Kaufhalle instead of Supermarkt („supermarket“) or Plaste for Plastik („plastics“). A few even managed to get adopted into the reunified German lexicon like Ampelmännchen (the little green man at traffic lights) or the Grüner Pfeil (a turn on red permission sign). But generally speaking the 40+ years of being apart didn’t manage to leave lasting traces in everyday language, and the few GDR words that remain aren’t recognised anymore as such, at least by the generations that grew up after reunification.
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The answer to your question was heavily discussed during the time of the separation of Germany, and it is perhaps not surprising that the answer is more difficult than one might think (and also different than what many German speakers who haven't looked at the literature would probably claim).
A recent summary of the research in this area is by Albrecht Plewnia,
https://ids-pub.bsz-bw.de/frontdoor/deliver/index/docId/4237/file/Plewnia_Deutsch_in_Ost_2015_1.pdf
which is well worth a read if you speak German. I'm trying to summarize his main points in the following.
First of all, because of the complex history of the region that we now call Germany, where a feeling of "Germanness" really only started to form in the 19th century, there are strong regional differences in the German language. In general, there was and is a strong gradient between the north and the south of Germany, and a smaller gradient between the east and the west. The differences between these dialects are very significant. See Fig. 2 in the article above for a good summary of the different language regions, and https://www.atlas-alltagssprache.de/ for an academic project to map current use of various words (which ignores dialects but illustrates the regional variety of a large number of common German words; if you do not speak German, click on "Erste Runde", "Zweite Runde" etc. on the left hand bar and then on one of the words, this will then show in maps of the usage of these words in mordern [post unification] German). While drawing analogies to other languages is always difficult, it is perhaps not too wrong if one argues that the difference between the northern German dialects and southern German dialects is similar to the variety of different accents of English on the British isles.
In other words: when studying the differences in German between East and West Germany (and also between Switzerland, Austria, and Germany) one needs to distinguish between the differences due to the dialects and the real evolution of the language due to the separation, e.g., of East and West Germany.
There is another complication, as Plewnia argues: While it is very easy to prepare long lists of words that were different between East and West Germany, a large number of these words come from the realm of bureaucracy - they are words that designate specific institutions or country specific processes. These are designations that are defined in laws, and are not indicative for a divergence in the language - think, for example, about the names of examinations: "Abitur" or "Matura" is used in the German language countries for the equivalent of the "A levels" or the SAT, but these different words are primarily due to the various procedures used in different areas to indicate a certain level of aptitude after school, but not really indicative of differences in the language. And the same can be used for the names of bureaucratic institutions that have similar tasks but are named differently (e.g., "Bundesversicherungsanstalt für Angestellte" (West) versus "Volkssolidarität" (East) for the Social Security Office / National Insurance). But, overall, most German words used on both sides of the wall were identical.
Correcting for all of these effects, Plewnia argues that most of the measurable differences in the usage of German between East and West Germany are just due to the historic differences in the dialects that existed already before the separation and then continued to evolve, and not due to the (not absolute) separation between the two countries. Similar differences can also be seen with respect to Swiss German and Austrian German: There is a continuum of dialects, which change (roughly) along the borders between the different regions of the German speaking countries, and in addition there is the sphere of "bureaucratic German" which is a consequence not of a natural divergence of the languages, but because of different bureaucratic approaches in public administration.
Now, having said all of this, there are indications that the wall was responsible for a slight divergence in the development of dialects, as measurable by subtle changes in pronunciation that were measured by comparing the language development in neighboring areas on both sides of the border. A summary of research in this area is given by Harnisch, Dialektentwicklung am Rande des eisernen Vorhangs, https://www.bpb.de/shop/zeitschriften/apuz/32953/dialektentwicklung-am-rande-des-eisernen-vorhangs/ (in German). For example, the pronunciation of the rolled "r" in the Coburg area and the Sonneberg area (separated by only a few miles) changed during the separation, probably because speakers in these areas were more influenced by speakers from other areas of Franconia and Thuringia, respectively. But these were very subtle changes.
German has a variety of dialects that might differ from accent to vocabulary, this alone is an extra difficulty Germans have to understand each other, however the Hochdeutsch (Standard German), spoken by most Germans, don't have as much differences as their dialects - which is not to say that there aren't general differences.
There were terms used specifically in each part, some of them fell out of use after the reunification, but others can still be heard today differentiating the German from people that were from the East Germany (GDR) and West Germany (FRG).
"Mit der Schwalbe zur Datsche – so sprach der Osten" is a book by Antje Baumann to shine some light on the differences of terms from GDR, such as the onomatopoeia for ducks: in FRG it was "quak, quak, quak", and in GDR was "nak, nak, nak", the latter being due to a well-known children’s TV character, Schnatterinchen the duck. There is also the Antifaschistischer Schutzwall (anti-facist protective wall), the GDR name for the Berlin Wall, to the widely spread phenomenon of the Westpaket (a care package sent by West Germans to the East).
Even the "Datsche" in the name of the book is a reference to a word used in GDR that has its origins in Russian, да́ча (datscha). Another cultural aspect absorbed from Russia was the different on the Kindertag (Children's Day), in FRG it was (and still is today in reunited Germany) September 20, but GDR had its Kindertag on June 1^st, much like how it is in Russia, and it was a bigger thing than in its Western's counterpart, specially because it was "a day off school spent singing songs, exchanging gifts and going to parties."
Talking about Russian influence, the GDR would also rather construct new words out of older versions of words, instead of adapting to Anglicised version, as the FRG did with its Americanised view. As an example the word for "fried chicken" in the FRG was Broiler, as in the GDR they used Brathähnchen. Another example is the popular saying Das hat Sinn, kept in the GDR, but the FRG changed it to Das macht Sinn, yet another example of the American influence on FRG.
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How the German language differed between East and West - translated from the original: Mit der Schwalbe zur Datsche: So sprach der Osten