Not sure that Salzburg would be that much cheaper - there are cheap(er) places in both cities. Unless you are talking about flights as Salzburg has some cheaper operators.
If you like culture and bigger cities, then I would say Vienna wins hands down.
Lots of interesting places to see and things to do.
Salzburg is smaller and more accessible and closer to the mountains. Also got the history and the culture, but a bit chocolate-box-y for me. Depending on when you go in October you might have lots of American coach tours in the place looking for Sound of Music sites. (Although they might have cancelled now wrt the world situation...)
I prefer Innsbruck to Salzburg as it seems more of a working city rather than just full of tourists. And if you are into mountains then they are on your doorstep.