Why was the Dual Monarchy of Austria/Hungary important in European history between 1867 and 1918?
I would say that the main importance of the Dual Monarchy during the time that it existed was the fact that its involvement in the Balkans led to the start of World War I.
The Dual Monarchy was, of course, led by monarchs who were ethnically German. However, it governed areas that had many different ethnicities. The most important of these (in terms of world history) were the Slavs in the Balkans. This was especially true after the 1908 Balkan Crisis in which the Dual Monarchy took Bosnia-Herzegovina over the protests of the (Slavic) Serbians.
The rising nationalism of the Slavs inside and outside of Austria-Hungary helped cause WWI when Slavic extremists assassinated Archduke Franz Ferdinand in 1914.