All-season tires are designed to work adequately in all conditions but excel in none. They're a compromise. In light winter conditions, they're fine; in serious snow, they're outclassed.
Winter tires (sometimes called 'snow tires') use rubber compounds that stay flexible below 40°F. The tread pattern bites snow and ice. Stopping distance on snow is 30–50% shorter than all-seasons.
If you live in a market with regular snow (more than a few storms per year), winter tires significantly reduce your risk of winter roadside calls — particularly winch-outs from sliding off the road.
If you live in a mild market with occasional snow, all-seasons are adequate. Switching to winter tires for one or two snow days isn't economically justified.
Storage: winter tires need a place to live in summer. Many people forget this and end up with damaged tires from improper storage.