Before We Begin
German is somewhat notorious for the extreme differences in various dialects, both in accent and in vocabulary. These dialects have been changing and evolving for hundreds of years. However, even in the last fifty years, one more difference has been created in the German language, that of the East varieties versus the West varieties. Here's an article covering how this split happened, the extent of the differences, and some common examples.
This article specifically refers to differences between East and West Hochdeutsch, the standard version of German that most Germans speak. Differences between the German dialects of different regions—Schwäbisch, Bayrisch, and so on—even if they are definitely located in more eastern or western (or northern or southern) locations are not the focus of this article.
Also, the differences between the dialects are far greater than those within Hochdeutsch: While East and West Hochdeutsch is mutually intelligible, if perhaps regarded as “quaint" by the other, many Germans have difficulty understanding each other's dialects.
One further disclaimer: These are general differences. Do not expect every specific East or West German to follow these trends. The particular attitudes and sentiments of the individual, especially with regard to political forces that shaped these differences, play a large part into whether they follow these trends or not.