Contrary to popular belief, Audi and BMW are not direct competitors to each other. In fact, BMW's only real competitor is Mercedes-Benz. Audi may build "powerhouses" but they tend to be very heavy with a much more stale ride quality in comparison. BMW makes Audi look either like Lexus or Maserati depending on the circumstance.
The RS models today barely beat the mid-range M models, most specifically M240i vs RS3, M340i vs RS4, M440i vs RS5, M550i vs RS6, M850i vs RS7, and even the new 760i vs S8.
Mercedes is alright but more of a car for Brad Pitt and Justin Bieber types, but BMW really provides the best bangs-for-bucks.
If you look back to the 1980s and especially the '70s, Audi's technological advancement and sporty enhancement both were closer to that of Peugeot, Renault, and Alfa Romeo (which back then actually wasn't a luxury brand at all).