Depends, where you live obviously. Yes we get the occasional flurries here in Cirencester, the pre-Christmas snow was the worst here in several years, just one day I didn't/wouldn't go out. Remember it's the other planks out there as well as you, regardless of you being nice and safe on snow tyres.
That does concern me. Though the grip from winter tyres increases safety margins significantly.
AWD enabled me to get in and out of my driveway something my FWD couldn't manage as well as my neighbours RWD Merc SLK!
Obviously having 4 driven wheels is an advantage when grip is limited.
So back to basics, what level of snowfall and how often your area gets it, dictates the expenditure and storage!
Especially if you have AWD which as stated is far better than 2WD in poor/any conditions (if it's a Golf that is)
Better at accelerating, I agree. However, little better at cornering and no better at stopping.
Having a car that can accelerate well is not safe if it cannot stop!
On snow and ice - a 4WD car will help get you up a hill, but having 4 driven wheels WILL NOT help going back down the hill. Going back down the hill the grip of your tyres is the most important factor.
Anyway, back on topic...
If you are trying to save money, I doubt if the purchase price of the 2 cars second-hand would be much different. The running costs are not significantly different either. Obviously fuel economy depends on journey type and driving style. However, for most drivers the difference always seems to be just around 5 mpg. Most people recommend running the Golf R on super unleaded. Some have said the GTI is less fuel fussy as the engine is not as highly tuned. The Golf R has higher road tax too, but probably less than you are currently paying on the older S3.
The GTI has a sportier style than the more reserved styling of the Golf R.
I know the Mk7 GTI is not in the same performance league as the Mk7 Golf R. But it isn't exactly slow and given the power-to-weight ratio it should be able to keep up with the older S3.