My view: the Golf R stands out more from a GTI or standard Golf, where the S3 doesn't stand out much sitting next to an A3 S-Line. I posted a photo of my A3 S-Line on Facebook and at least 4 or 5 comments were something along the lines of - "Awesome - nice S3" or "Is that the 300hp one?". Those people commenting were not car people...or were BMW/Merc people...but it just shows they couldn't tell the difference.
The same "problem" exists with the BMW 1-Series - I honestly can't tell between an M135i and a BMW 1-Series with M trim - I'd have to look a the exhausts to know. The A45 AMG does stand out but even then if you throw on the Night Pack and AMG Styling Pack onto a standard A-Class then it starts to look very similar.
I also think the Golf R has more boy racer looks, where the S3 is understated and classier. For example I'd happily drive an S3 to a posh dinner party and park outside the entrance - proud of it. But I'd probably park the Golf R further away. Those rims and pipes look after-market.
I'm also a self-admitted badge snob, so despite the more aggressive styling of a Golf R, I'd still have the S3 because it's Audi and it's something to be proud of in any occasion.
The interior makes a big difference as well. I sat in GT-R a few years ago and it's jaw dropping on the outside but the interior is nasty - feels cheap despite the ridiculous power. I get the same impression with a Golf - yes they share the same engine parts etc. but the A3/S3 interior impresses most passengers. The pop-up screen, cleaner centre console, feel of materials, the way the dials click and the glovebox shuts etc.
In that respect I think the S3 (or even A3 if you're comparing against a Golf) is the best all-round package.