There is a very weak definition of what an M+S tyre actually is.
It is something like, a tyre that is specifically designed for Mud & Snow….
Bit like comparing a supermarket mountain 'style' bike with a 'real' one.
Basically it is up to the tyre manufacturers if they 'believe' their tyre is suitable for such a marking. There are definitely some tyres marked as M&S that have rather limited winter performance. Winter performance comes from a combination of pattern, compound & construction whereas a mud tyre has less reliance on tread compound.
The snowflake marking is a US Federal marking. This means the tyre must pass a snow traction test to receive the marking though this is up to the manufacturer to confirm. As the penalties for 'being mistaken' (or lying) are rather high, if a tyre has the snowflake marking it can be normally considered as valid but this is only truly reliable in the US where the mark has some value. In Europe there is no test required to display the marking.
There are tyres available that are known as 4-season or all-season pattern. This means that they somewhere halfway between a summer & continental winter tyre. The typical snow performance for such a tyre means that passing the snowflake mark test is rather borderine.