Hiya mate..
Hopefully i can answer this one for you as i had a 2006 MKV GTI before my current A3 2.0TDI...
Firstly, the GTI is a fantastic drivers car and the 2.0TFSI engine is a cracking bit of kit imo, but once remapped the 2.0TDI is not 'that' far off in performance terms and i have never felt 'short-changed' in the performance stakes since changing cars..
Handling is ultimately better than my 06 S-line but nothing that a vagcom change to the steering boost and a rear ARB wouldnt sort out imo (i'd personally put the A3's handling at 90% of that of the GTI if i had to quantify it with a figure..)
Braking is definately better on the GTI with the use of 312mm front versus 288mm on the TDI, but neither cars have what i would call great brakes and i always lacked confidence in the GTI setup (the A3 brakes just plain scare me lol!)
Where the A3 easily scores points against the GTI imo is in the build quality area, my brand new GTI rattled and squeaked like you would not believe and the stuffing was gradually coming out of the drivers seat and the dash trim and plastics quality is a world apart from the A3, very cheap and grey looking and i never really felt that i was sat in a £21K car if i'm honest..
Looks wise i much prefer the A3 as i always felt the Golf to have too high a roofline and a fat ****, but it's still a great looking car and VW got the GTI styling just right even if you felt like you had to wear a top hat to fill the roofspace lol
Running costs wise the GTI over 10K miles averaged 29Mpg overall and would return 34ish on the motorway and around 25 around town, full tank range of between 300-340 miles and £700 pa insurance..
The A3 over the same mileage has averaged 47Mpg overall, with 55ish on the motorway and 45 around town, full tank range of between 500-550 miles and £390 pa insurance..
In summary my A3 provides 90% of the driving ability of my old GTI but imo is a far nicer car to both look at and sit in and as yet has no rattles or squeaks and it cost £500 less than the GTI...
All the above long ramblings are of course only my opinion and i await the flaming from both the diesel and serial Audi haters on here to start...