For many drivers, all-season tires are standard fare on today's vehicles. These tires are adequate for driving in most conditions, including light snow. In heavier snow, most all-season tires can be overwhelmed easily.
I liken driving all-season tires to wearing tennis shoes; they will get you by in most situations, but have a limit. Imagine walking to your car across a dry parking lot in tennis shoes; no problem. Now add an inch or two of snow and things start to change. Up that snow total to 8" and the Nikes have lost their luster, and a pair of boots is starting to sound good. The soles that grip and create the familiar squeak on the basketball hardwood fill with snow, and act like skis in the severe winter conditions.
Tires are the same way. All-season tires allow the driver to use a single set of tires all year, but fall short in severe winter conditions. All-Weather tires, on the other hand carry the Severe Service Symbol:
Tires with this 'snowflake on the mountain' symbol on the sidewall exceed industry and government standards for use in extreme winter conditions. Historically, this rating has been given only to dedicated winter tires, but brands such as Nokian and Vredestein have 4-season tires that carry the Snowflake.
Vredestein Quatrac 3
These tires allow the driver to use a single set of tires all year long, while delivering outstanding winter performance.
Here is an interesting read on all-weather tires from the Canadian newspaper The Globe and Mail:
All-Weather Tires--The Globe and Mail