The lines most affected were U6 and U8, which continued to run under East Berlin without stopping, through “ghost stations,” where East German soldiers stood guard on dimly lit platforms. A special case was the stop at Friedrichstrasse on U6, where trains continued to stop for transfers to the S-Bahn. The DDR set up “Intershops,” where duty-free goods were sold at prices much lower than in West Berlin. The shops became so popular—expanding from tobacco, alcohol, and candy to perfume, toys, and records—that the stop became known as Kaufhaus Friedrichstraße. Here’s a map showing the ghost stations.

During the decades of separation, both West and East built expansions to the network. One interesting difference seen in the East is that stations there commonly have single stairways in the center median of a street, descending directly to the platform, rather than stairways down from sidewalks. Apparently vehicular traffic in the East was never great enough to present a problem with the arrangement. In West Berlin, by contrast, the growth in vehicular traffic presented operational problems for the city's trams, and they were all removed. East Berlin, like most Eastern Bloc cities, kept its network of trams.

This topic has been of considerable interest, not only to railfans. Brian Hardy’s book The Berlin U-Bahn has a succinct description of operational challenges, as does this Wikipedia entry.