In terms of classical music, the most significant impact of German unification was probably the decline of Vienna as a European center of power. Many composers from all over the German-speaking territories had agitated for national unification, and when the ultimate resolution turned out to be a Kleindeutschland Lösung (Little Germany solution, led by Prussia and excluding Austria) rather than a Großdeutschland Lösung (Greater Germany solution, including and led by Austria), this also weakened Vienna's musical hegemony over the German-speaking world, starting after its defeat in the Austro-Prussian War of 1866. In general, however, it seems that the Austria/Prussia schism was much less significant for German composers than other conflicts, like that between the progressives and the conservatives (see War of the Romantics). The German nationalist sentiment of the Romantic period continued in a somewhat different form after unification, having the full blessing of the autocratic Prussian-dominated state, but since the primary objective of national unification had already been achieved, artists refocused their goals away from clamoring for German unity.

One interesting story is that of King Ludwig II of Bavaria, who plays a unique role in the history of both German unification and German music. In 1864, having just been crowned, Ludwig invited the perpetually cash-strapped Wagner to his court at Munich, being highly enamored of his music (and apparently of the composer himself to boot). Here Wagner wrote Tristan und Isolde, Die Meistersinger von Nürnberg, and roughly the first half of the Ring cycle. Bavaria, meanwhile, sided with Austria in the Austro-Prussian War (both states were predominantly Catholic) and ultimately saw its fortunes decline in the face of Prussian domination. In 1870, Ludwig wrote a letter known as the Kaiserbrief which endorsed Bismarck's creation of the German Empire; this was done in exchange for secret payments which Ludwig used in the construction of Neuschwanstein, his gorgeously over-the-top castle. This castle was partly inspired by the king's fondness for Wagner and his music, particularly the medievalist nostalgia of Tannhäuser and Lohengrin. Relations between the two men later soured, and Wagner left court to build his long-dreamed-of opera house in Bayreuth, but that too ended up being funded largely by Ludwig. Ludwig more or less retired from public life after the humiliating partial loss of sovereignty to the German Empire, spending his time and money on other eccentric castle-building projects.

I'd recommend checking out Music and German National Identity (ed. Celia Applegate & Pamela Potter) and Imagined Germany: Richard Wagner's National Utopia (Hannu Salmi).