I cross shopped an S3 when I totaled my 2015 R. It was in between the first and second allocations of the 2016 R, so I couldn't find what I wanted, and you had to deal with the VW dealers treating it like it's gold plated.
Audi dealer totally willing to negotiate, and test drive. I actually really wanted to like the car but these were the issues I had with it.
Might be the same engine and same transmission, and platform, but they don't drive the same at all. Like all the reviews I've seen comparing the 2, the R just has a more nimble feel to it, the S3 just seems to drive "heavier". DSG shift points are different, and the magnetic ride, I don't think is as sporty as the DCC.
No back seat, I can fit in the back set of my R with no issues, plenty of headroom, leg room. In the S3, because the roof slants down so quickly, I hit my head on the ceiling, wouldn't want to ride in the back for very long. Obviously no hatchback (In the US), which I like the utility of, and it's almost a deal breaker for me.
LED nor HID headlights have adaptive lighting, I got used to lights turning into the corners when driving the R.
Interior is not my taste, other people thing it's an upgrade over the R, I disagree, it does feel more open, I like the cockpit feel of the R. Never liked round vents, and I prefer touchscreen over the touchdial thingy.
The R seats fit me much better, they really feel like a wraparound seat to me. I tried sitting on both the normal, and the high end sport seat. Neither fit me as well, or made me feel like it was going to hold me as well in corners.
The other issue with the S3, it literally looks like every other Audi sedan, I had trouble telling them apart in the showroom. In the end, I liked the car, but it didn't wow me like the R did, and it was still more money. Getting back into a 2016 R, was like coming home, and I didn't misremember how the car felt to me, vs the S3. I just got lucky that there was a second allocation.