I write this post every year because it's important. People ignore their tyres (the same way they ignore oil changes) but when piloting a two-ton weapon, the only control you have over the thing is the four dinner-plate-sized contact patches between the tyres and the road. Ignore the tyres and any hope of controlling the car goes out the window. Summer tyres are just that - warm weather, summer tyres. They are completely useless in the winter. Sure you might think they feel OK but when the time comes that you really need them - an emergency stop or evasive action - trust me - they'll be utterly useless.
All-season tyres are a bit of a misnomer. The tyre industry understood long ago that people ignore their tyres, and the percentage of people who actually care enough to change from summer to winter tyres is pitiful. So they came up with 'all-season' tyres. These are not really all-season tyres because they're a horrible compromise. They're bad summer tyres, and bad winter tyres. In fact if you want to be that lazy, just go ahead and put summer tyres on because at least then you've got reasonable rubber under the car for 8 months of the year. As oppose to all-season tyres where you have a piss-poor compromise all-year round.
Winter tyres are what you should be using in cold weather. In fact I would prefer the whole summer/winter thing to be changed to 'warm weather' and 'cold weather' tyres. Cold weather tyres have two very important changes compared to warm weather tyres. They're made of a much softer rubber compound that remains flexible to much lower temperatures, and the tread is designed offer a much better chance at gripping on snow and ice.
Yes yes I know - you're reading this and thinking 'but it doesn't snow where I live - it's just cold'. Doesn't matter. If the air temperature where you are is below 7°C in the winter, then the road temperature is going to be below 2°C and you need cold weather tyres. Summer and all-season tyres become rock hard at those temperatures and coupled with warm-weather tread patterns, they're largely useless in cold weather.
Remember - it's not the acceleration you need to be worried about. It's stopping and turning on cold, or icy and snowy roads.
The one caveat with cold weather tyres - once the average temperatures go above about 12°C, get them off and put the warm season tyres back on. Otherwise the warmer road surface temps will eat the super soft rubber compound on the cold weather tyres for breakfast and you'll end up with no tread left.
If you want all that summed up in a video - check this out, done with trikes, instead of cars, so you can see what a difference winter tyres really make: