Since the pure full blown winter tire and the 4 season all weather tire both have the snowflake/mountain icon, how does one tell the difference between the two types of tire?
The mountain snowflake is almost irrelevant because the standard is 10% better traction than a reference all season. Virtually every all terrain or 4 season tire produced today can crush that standard.
In this case, the marketing designations are differentiate effectively into what is really now “all weather” and “winter” classifications.
Within “all weather” you have two primary tire types: 1) what used to be all terrain but is now a hybrid offroad true four season tire in terms of siping, tread design, and compound, i.e. truck tires where the traction of “all terrain” includes rain and snow, and 2) tires that are more road biased blend like the Michelin LTX but retain those advanced all weather compounds with more siping. Nokian is an outlier with the WR series that nods to four season compound wear with a winter tire tread. Within the two types here, you are biasing to offroad use if you have it along with deep snow, or just somewhat lighter snow and all road miles (or more likely for most, looks).
Within “winter” there is deeper tread for snow vs. ice focus, with studs taking up most of the true ice spectrum, noting that there are studdable all weather tires above, but it really defeats the point of a four season tire to stud it. An ice focused tire is likely to be relatively (to other winter options) poor as snow depth increases and possibly in rain, which is the Achilles heel of winter tires generally outside of heat. These are all specialist tires that you wouldn’t run year round.
Comparing the two categories, the aggressive all weather tire with deep tread, inner tread channeling, and large outer lugs will outperform a winter tire in heavy slush, deep snow, and rain with the winter tire outperforming as conditions get harder packed and icier. There are some exceptions - the Nokian Hakka SUV is a more aggressive tire than a Michelin LTX and would probably outperform it in deep snow, but the logic generally will hold. On dry pavement it depends - the all weather tire will generally outperform both in tread life and traction and both will have some noise.
The biggest distinction at this point to me is whether you drive a truck or a car. The all weather tires are so good now blending traction and treadwear that you only need more raw tread traction to drive at unsafe speeds, but these tires are generally too heavy duty for cars. Cars also need a lot more forward traction assistance than trucks and don’t have the clearance or drivetrain to play in deep stuff or offroad, so the benefit of that type of tire is low.
Simply said, in my view a “winter tire” is unnecessary on a truck and will cost a lot more over the life of two tire sets than an all weather tire while restricting versatility in the process. And an all weather tire isn’t really designed for light duty cars and isn’t likely enough for 2wd in icy or steep conditions anyway, so with cars you are largely forced to still pay the piper.