Having driven a M135i today and sat in the S3 (couldn't drive it today) this clearly isn't as comprehensive as Paul's comparison but nonetheless I did form an opinion (and not the one I expected)
Surprisingly I warmed more to the M135i interior than the S3 in the flesh than I had comparing pictures. The Audi is a little austere in comparison and I wasn't so keen on the S3 dials as I though I would be - then again I am possibly the wrong side of the target age group
Practicality wise the two seem on a par.
Once I got over putting my foot down to hear the engine sound in the BMW I tried it at a sedate pace and the brilliant auto box made it easy plus the ride was compliant. This is a very capable car at driving around in congested traffic and ferrying the family around wihilst providing sports car type performance when needed. Something I thought might be an issue.
In the south west UK we get little snow and certainly had more in my youth in spite of what we hear about climate change. In 33 years of driving I have got stuck in snow in 2WD cars twice - ironically an Audi and a BMW! So 4WD doesn't feel a significant factor but other regions will vary and the Audi has a clear advantage in such a case.
Without driving the S3 I was able to compare the bits I expcted the S3 to win me over - how it made me feel sat in it and looking at it. The fact it didn't win me over is simply down to a preference thing. I would be extremely happy sat in and owning either car, I think they are both fantastic cars.
If 4WD is important the Audi is a clear winner. If the chance to own a new small high quality hatchback with a 6 cylinder engine and RWD before the next generation BMW 1 becomes 4 cylinder and FWD matters then it has to be the BMW. I probably fall into the latter. I owned one of the last of the V6 Alfas before they were replaced by the GM variant and as much as I admire the engine implanted into the S3 the opportunity to enjoy that BMW 6 cylinder engine while it is an option may be too much to resist.
The counter of course is that small BMWs of the future will be to a format that Audi are building now, so the Audi is the more modern technology package whereas the BMW appeals possibly more to those looking for more old school driving thrills.
I would be very surprised if buyers would regret owning either car but that doesn't make the choice easy