Erm.....
en.wikipedia.org
Both nations historically spoke the same language (German), so in that sense sometimes an Austrian (in many cases, a Viennese) composer might be regarded as German. The German nation we know today and who fought France and England in both world wars was originally Prussia and several other small German-speaking states located in northern European east of France. Austria was another German speaking confederation of lands that later became the Austro-Hungarian Empire and basically controlled the southern half of central/eastern Europe, extending from Prague and Cracow in the North to the Adriatic Sea in the South. The Hapsburgs controlled it for the most part. The term 'German' when used to describe music likely refers to the language, regardless of whether it was a composer working in Berlin (Germany) or Vienna (Austria).
Keep in mind that Austria-Hungary was the country that "started" the First World War, when a Serbian radical killed Austria's arch-duke. The entire so-called 'powder keg' was within Austrian borders. After WWII, Austria shrank down to less than half the size of the H-A empire, splitting into what would later be called Yugoslavia, Poland, Czechoslovakia, Hungary, and German Austria. When both wars were declared, Austria and Germany were allied (before the second world war, Germans simply walked into Austria and annexed it back into a 'Greater German nation'). In both wars, the Germans certainly fought more voraciously, since they were actively industrialized and spurred on by imperialist propaganda. Both times, Austria was simply an old-money tag-along who were likely after a piece of their former Hapsburg glory.
Topic: What's the difference between Germany and Austria?