What are the rules?
The rules are given in the DMG (page 110) as follows:
EXTREME COLD Whenever the temperature is at or below 0 degrees Fahrenheit, a creature exposed to the cold must succeed on a DC 10 constitution saving throw at the end of each hour or gain one level of exhaustion. Creatures with resistance or immunity to cold damage automatically succeed on the saving throw, as do creatures wearing cold weather gear (thick coats, gloves, and the like) and creatures naturally adapted to cold climates.
Am I missing something?
Technically, you are not missing anything. Per rules as written, if the characters are wearing cold weather gear in freezing climates, they will be fully protected.
Overall in 5e rules that deal with logistics and resource management have been de-emphasized compared to early versions of the game. For example, the default rule for carrying capacity is a very generous 15 pounds of weight per point of strength without any adverse effects on mobility, and the PHB says on page 178 "which is high enough that most characters don’t usually have to worry about it." Even a commoner could move without impediment carrying 150 pounds. That same commoner would only need to eat every four days to stay healthy.
Neither of these rules is realistic. As is often stated, D&D is a game, not a physics simulation. From a gameplay perspective, what rules about carrying capacity, water, food, torches and weather protection often amount to in practice is bookkeeping. Some people enjoy that aspect of the game, others do so less. The rules of 5e make it relatively easy to largely ignore that aspect of the game, if you are not interested in it. (They provide a few tools, such as Variant: Encumberance rules on page 176 of the PHB, if you are).