I've had both (although my A3 was a cabriolet).
The A3 was considerably nicer than the Astra inside, and the seats were more comfortable on long journeys.
The Astra was more spacious on balance (factoring in that the Audi was a convertible, so the boot space was configured differently). The boot in the Astra was huge.
The gearbox and handling were much better in the Audi, but then the Astra was a diesel so nose-heavy (diesel engines are generally heavier than petrol).
My A3 had the 1.4 TFSI engine with cylinder-on-demand and it was very economical.
Audi main dealership servicing costs a lot. If I remember correctly a major service was about £400.
But that goes for most makes really, even Vauxhall servicing isn't especially cheap these days.
Either way, you'd probably get those costs down a bit if you serviced at a well-regarded independent garage rather than a main franchised dealership. (Some things will always be more expensive though, certain parts for example.)
Mazda 3, Volvo V40, Toyota Auris or Kia Ceed/Hyundai i30 great alternatives to A3 or Astra.