One can consider the Serbo-Croatian language. Due to the differing histories, Croatia is largely Roman Catholic and was under the influence of the Roman Catholic Church, while Serbia was Orthodox, and fell under the influence first of the Byzantine Empire, and later on the Russian Empire/Soviet Union. Serbian and Croatian are essentially the same language, but Serbian generally uses a Cyrillic alphabet, while Croatian uses a Latin alphabet. There are other minor differences, but the alphabet/writing system are the biggest divide between the language in Croatia and the language in Serbia.

The best example I can think of is India and Pakistan, where the language of Hindi in India developed into Urdu in Pakistan and was solidified after the partition of the two countries occurred.

In regards to the spoken language the differences are subtle, and generally the only times you'll notice are in more formal programs such as news broadcasts in either country.

The written languages on the other hand are completely different. Urdu uses an alphabet consisting largely of arabic letters, while sanskrit is a Devanagari script.

What's interesting is the countries have only parted roughly 60 years ago, yet people from either country could most likely not read the other's written language.

For example the Nordic languages in Scandinavia and Denmark - it was once a big dialect continuum, but after the formation of the stats each one gradually switched to the language of its respective capital. This was the case of many languages across the world, another examples include Indonesian and Malay, more recently Croatian and Serbian...it's one of the basic ways new languages are formed.

One very interesting example that comes to mind is that of Dungan. There's a fascinating if somewhat outdated piece by Victor Mair explaining the ways in which Dungan has diverged from other Sinitic languages by virtue of its rather unique orthography.

One of the most readily evident divergences (extensively discussed at the above link) is the way foreign words are adapted to Modern Standard Mandarin and Dungan. As Mair notes, MSM has fairly strict demands on the sound-shapes of words, due in no small part to the demands of Chinese characters, so words with consonant clusters are drastically changed in a variety of ways. Chinese characters used to represent borrowings often have their own meanings, sometimes consonant with the borrowed word (Mair gives by way of example ""miniskirt" → miniqun ["skirt which entrances you"], "husband" → heiqibandeng ["black lacquer board stool"]" and a few others). Borrowed words are often clipped to a single syllable, obscuring their origins, and a single borrowed word can be adapted to MSM in many different ways. Finally, borrowing is often simply discouraged in favor of new coinages.

Dungan, on the other hand, developed an alphabetic script under Soviet control, and so the strategies used to adapt loanwords to MSM are simply not necessary.

This is simply one of several effects Soviet rule had on Dungan. I'd encourage readers to check out Mair's piece.

It would be interesting to know if/how this developed in west/east Germany and particularly in what way the two dialects converged again after the reunion