I would go with the Mazda3 hands down. The interior of the new one is astounding for the price, competing favorably to even some new Audis. The A4 is an old platform dating back to 2007. Really should feel older than the Mazda. The interior of the Audi is super old and the Mazda's is brand new.
Mazda made a mistake with the torsion beam rear suspension. At least some reviewers are saying they wouldn't have guessed it was a torsion beam setup unless they were told so. Also, they still have a lot of their 6 speed autos in stock, so they chose to use what transmissions they have in order to save money. Some steps in the wrong direction, but ultimately they succeeded with the interior and premium feel. Fuel economy falls behind some of the competition as well.
These are both compact cars, and they aren't that roomy inside compared to other compact cars. If you want more space, I suggest looking at a fully loaded Civic Touring or Sport Touring Hatch.The Civic does everything well, but the Mazda excels at feeling like you got a lot more than you payed for. I would have to drive the Civic and Mazda, but I would choose the interior of the Mazda with the dynamics of the Civic or the previous Mazda3. I have sat in the interior of the new Mazda3 and it would beat the Audi hands down.
I think after driving the Mazda you would notice the Audi having more low-end torque since it has a turbo. You may have to rev the Mazda more often.
After noting that reliability in the Mazda will probably be a lot better, the Mazda is the no-brainer choice. I would argue the Mazda is the more luxurious.
Please let us know what you decide on.