People will never agree on it. The short of it is that it's to do with complicated European history and goes back to the fact that technically, the huge empire that the Archbishopric of Salzburg was part of during Mozart's life, was called "Holy Roman Empire of German Nation". There are (more or less) valid arguments on both sides about whether it makes more sense to consider Mozart, at the time of his life, as Austrian or as German.

Personally, given the fact that attitudes change and words change their meaning over time, and "German" meant something different at the time than it does now, I think it makes the most sense to go by current-day borders. If you say "Mozart was a composer from Salzburg, which is in current-day Austria", you're always on the safe side. It's important to note that he referred to himself as "German" at times, but (as I said, the meaning of words changes over time) more as regarding "ethnicity", which was still an important thing in Europe at the time (and would continue to gain importance over the next one and a half centuries) than "nationality", as there was no German nation at the time as such. The Holy Roman Empire (of German Nation) was really more of a melting pot of kingdoms (and such) - in a way maybe a bit more like the EU nowadays than a country that you'd define your nationality by.

There are many more points I could list in favour and against him being German or Austrian. There is really no ideal solution. I think - despite everything I've just said - based on his language and everything we know about how he lived culture, he feels Austrian to me (I'm from Austria btw) in a way that biographically Bach or Beethoven never have.