The difference between the Audi a3 and s3 is in the body design,
and engine, for the most part.
"S" models from Audi are always more powerful and tuned to be
sporty than their "A" counterparts. As such, they are more
expensive.
The current generation Audi S3 (2006 to present) has a 260hp
engine, vs 197hp in the A3's 2.0L turbo engine. The engine features
an upgraded turbocharger, pistons, intercooler and more. The
four-cyliner engine is good for a 0-60mph time of around 5.5sec
(that's fast! The A3 does it in around 6.6 sec), and a top speed of
155mph.
Also, the suspension is tuned giving the S3 a lower appearance
with stiffer suspension, wider tires, and better handling.
Various body moldings, spoilers, and bumper covers are upgraded
to look more sporty for the S3 Audi.