My Mk6 Golf GTD comes with 17 inch alloys as standard. The workshop manual actually recommends 16 inch tyres for winter driving. I have a second-hand set of 16 inch alloys for cold weather tyres. (I don't think I will be getting winter alloys for my next car.)
Advantages: Smaller tyres are cheaper. Narrower tyres have better PSI road contact. Therefore, they perform better on snow and ice, and are less likely to aquaplane. I can change the tyres anytime I want. 16 inch tyres are slightly more comfortable and quieter than 17's. Don't care (much) if I hit a pothole or kerb.
Disadvantages: My winter alloys look like poo on the car. Good alloys can be expensive. I have to fit them myself.
My wife does not have a second set of alloys. She gets her winter tyres remounted every year. If you want to do some research on this method, google search 'remounting tires.'
Advantages: Do not have to buy another set of alloys. I do not have to change the alloys. The car keeps the same appearance as the same alloys are used for summer and winter tyres. The tyres get balance regularly.
Disadvantages: Cost of tyre storage bags £25. Cost of getting tyres remounted (£25 each time.) Some tyre fitters will agree a price for remounting. Especially if you buy the tyres from them. Have to find time to take the tyres to be fitted. As tyres are change on the alloys twice a year - there is a greater chance that the alloys/tyres could be damaged by the fitter.
I much prefer my wife's method of fitting winter tyres. The tyre storage bags have handles and they keep me and her car clean.