The weight of the S3 over an EP3 alone would make a huge negative difference, so you're comparing a more focused, lighter car to a pretty porky one that's focused at a slightly different segment and one that, out of the box, doesn't have that much of a power to weight advantage.
What the S3 has is way more grip than you would need at anything below Stage 3 (hybrid or big turbo power levels) and that, budget allowing, allows for a very potent car.
@DJ_26 is spot-on as far as handling mods go. One thing I would add is suspension and engine mount bush upgrades go a long way to making it come alive when at the limit. I only have the uprated dogbone mount insert and that has made a noticeable difference to how the power is delivered.
One of the biggest weaknesses of a stock S3 are the front brakes - OEM kit is completely inadequate for the weight of the car and you can easily run out of braking ability during a spirited drive on a back road, let alone on a track. Again, many options here to fit different budgets, but cheapest and most effective mod would be uprated disks, pads and braided lines all round.
All these mods cost a significant amount of money, so what your decide ultimately will depend on what you want more out of a car and your budget. An EP3 is more fun out of the box on a dry sunny day, but the S3 is a well-rounded car that can eat up miles easily, stay civilised, but can also can be made to surprise much more 'sporty' cars if your pocket allows. The look on one guy's face the other week was priceless, after he got very embarrassed in his brand new Focus RS three times by a 9 year old shopping trolley
