I've owned the last iteration of the S3 (3 years from new) and then traded that for a TTS (2 years from new), so here's my TPW.
Costs? My 8P S3 was a manual and the TTS an S tronic. I only mention that as the S tronic in that car used more fuel. Running costs? Not the same! Far from it! The TTS, as stated, used a lot more fuel, especially when you drove it hard. I used to average around 33mpg in the S3, in the TTS, more like 28. Insurance, the TTS was around 25% more expensive. Tyres? I had 19" alloys on the TTS, but then the standard fit 18's for that car are still shod with wider rubber than the S3's 18 inch rims. So, you're looking at around a grand for tyres and just over half that for the S3. Servicing, the TTS was again more expensive, which always struck me as odd, given it's pretty much the same car underneath. To buy, well the TTS will cost you more but the S3 held it's value batter when I traded. In fact, it retained 60% after three years and 45k, versus the TTS at 55% after 2years and 58k.
Problems? Not many really. Squeaky brakes - yep, had that on both cars and in both cases, rectified by twisting Audi's arm to replace the pads under warranty I would add! A really bad misfire on the S3 was a problem that Audi took ages to fix for me (four times it went in before they fixed it). Rattling windows on the S3 was a constant issue until I discovered how to fix it (Audi never managed to sort it out). With the TTS I found the tracking would get knocked out quite a lot, never really sure why the S3 didn't suffer the same problem...
To drive? The mag ride on the TTS made the handling very inconsistent and IMO spoilt the car a bit. The TTS was the faster car of the two and made the better noises, but the S3 handled better, probably something to do with the mag ride? I dunno, but I had more confidence with the S3 and its ride was way better. The S3 attracted far less attention on the road and the TTS definitely promoted the green eyed monster in people - it was after all a very nice looking car (mine was a black edition in glacier white with titanium finish alloys). I used to find **** heads trying to race me all the time, people not letting you out in traffic, etc. Never really got that when in the S3.
Day to day use? The TTS was surprisingly practical if you don't of course need the rear seats. I used to get all my fishing gear in, no problems at all. The S3 actually has quite a small boot and even with the seats folded, actually has hardly any more space than the TTS, albeit, in the TTS you lose height space, if that makes sense. There was also the Elephant in the Room with the TTS - hairdresser jokes and such like did bore me a bit and it was a tad boring to drive.
Conclusion? Both great cars, but IMO, the S3 edges it. It is cheaper to run, almost as fast, a little more practical (you can have passengers!) and is subtle in a good way. Don't get me wrong, I loved my TTS, but it didn't rock my boat like the S3 did. The S3 was simply better to drive, but I suppose on reflection, didn't always feel a special as the TTS. But that wasn't really too important to me. I went on to get a TTRS, which I would say was a bit of a mistake, as the premium I paid over the TTS was hard to justify for what you got. Now in a new S3 and happy again... hope that helps.