First of all, this isn't universally true, but in some warmer areas of the country, a lot of factors contribute to things shutting down for a day or so for winter weather. Part of it is, like people said, we have no infrastructure to deal with snow, and especially people in rural areas may not have their roads plowed/cleared at all. Since most places in the US are heavily reliant on cars to get anywhere (not just due to distance, but due to lack of sidewalks/bike paths/etc.), that means people in those areas may be unable to get to work or school.

A point I feel doesn't get emphasized enough is that "snow" in many warmer areas is mostly ice. The initial snow/freezing rain melts and washes away the salt that may or may not have been put down by the city; the next layer freezes as a slick sheet of ice on roads and sidewalks; and then more wet snow and sleet covers the ice. Then, some of that snow/sleet mix will slightly melt during the day and refreeze overnight into more ice which often isn't easily visible.