Snow tires (as in Mud and Snow) were a particular breed of tires that had better traction in mud and snow, which were considered to be equivalent low traction situations requiring special tread patterns. Sometimes a special tread compound was used - walnut shells being one interesting ingredient sometimes used.
Snow tires worked fairly well compared to the regular tires at the time, which was before radials were introduced. Typically the cars were rear wheel drive and Snow tires were mounted on the drive tires. It was known that the steer tires could use the additional traction, but overall, snow was such a low traction situation, it hardly mattered.
When radial tires were introduced, the improvement in traction was quite apparent, and because the wear rate of radial tires was much better, the tread pattern could be more aggressive without losing too much wear - hence the birth of "All Season".
Just about this time, Front Wheel Drive was introduced. It took 20 years, eventually FWD dominated the car market.
What was interesting was that the California Highway Patrol required "Mud and Snow" tires for certain weather conditions, so a definition for "All Season" tires was developed by the Rubber Manufacturers Association (RMA). This definition described the appearance of the tread pattern, and if the tread pattern met that definition, the tire could be labeled with the letters "M" and "S" with an additional symbol between the 2 letters, such as "-", "/", "&", or "+".
Why a description of the tread pattern, rather than a test? Because, snow traction testing was in its infancy and wasn't reliable enough.
Needless to say this created confusion because "All Season" and "M*S" have different letters and the concepts are quite different. Over time, the use of snow tires reduced to the point where it was hard to find snow tires.
Over in Europe, they went a different direction and created "Summer" tires and "Winter" tires with each designed for the specific conditions. These tires were quite different both tread pattern-wise and tread compound-wise.
The Europeans noted that radial Winter tires had so much better traction than bias Snow tires that the use of 4 Winter tires became more or less standard. Not to mention that FWD cars would lose control at the rear with winter tires on the drive axle, which is not the way you want the car to behave.
After a while it became apparent even in the US that Winter tires had much better traction than All Season tires and Winter tires started to sell. By this time, winter tires had much much better traction than the old bias ply snow tires that marketing the tires as "Winter" as a good idea to emphasize the improvement.
In some locales (Quebec comes to mind), the improvement in traction of winter tires was such that they felt the need to mandate winter tires. The problem was determining what was a "Winter" tire. The Canadian government asked the RMA for help and the RMA came up with a standardized test and the "Mountain/Snowflake" symbol to indicate passing the test. This was intended as a winter tire test, but it was quickly discovered that some All Season tires and most All Terrain tires would pass the test.
However the Canadian government wanted the symbol to delineate winter tires, not aggressive all season tires, so they asked the RMA to mandate the symbol for use only on winter tires - which the RMA can not do, but they passed that suggestion onto the tire manufacturers, who aren't always following that suggestion.
Meanwhile, the Europeans suffered a bout of "Not Invented Here", so the symbol is NOT looked on with favor in Europe and some winter tires over there do not have the symbol.
So, today, we have a mix of things. Winter tires sold in Canada and the US have the Mountain/Snowflake symbol, but not necessarily in Europe. The symbol appears on some all season tires and some all terrain tires, but absence of that symbol does NOT mean the tire will not pass the test.