The term "mud and snow" can cause confusion when talking about tires. If a tire is mud and snow rated, it will carry the "M+S" or "M&S" branding. But what does that really mean? Is a tire with the "M+S" branding intended mostly for mud? For the snow? Off-road?
The mud and snow branding is really a very general indication that a tire is built to handle some bad weather. Any tire conforming to some very basic requirements for void area and positioning of grooves in the tread pattern can be branded "M&S" at the manufacturers discretion.
In theory, even most summer-only performance tires could qualify for the mud and snow branding, but in the interest of reducing confusion, tire manufacturers generally only use this mark on tires that are intended to offer at least some level of winter capability. Despite the name, most mud and snow branded tires are not specialists in mud nor snow.
If you need a tire for serious muddy conditions, look into purchasing a specialized mud tire. There is no special branding to indicate a mud tire, but they are easily identified by their large, widely spaced tread blocks. They're marketed for off-road use and their names will often include references to mud or off-roading.
When looking for a tire specifically for snow use, look for the mountain/snowflake branding that indicates a tire is truly designed to excel in winter conditions. Should you need help identifying the sidewall markings on your tires, read "Sidewall Markings."