Often ads or tv spots make it look like succulents can thrive anywhere in your house. Sadly, that's not quite true. Most succulents need a substantial amount of light to stay colorful and compact.
Many succulents are native to temperate areas of South Africa. As such, many succulents have a hard time when the weather gets extreme -- either cold or hot.
Cactus are a distinct group of plants on their own, but they're actually part of the succulent family. All cacti are succulents but not all succulents are cacti. It's not a surprise these prickly plants fit into the succulent family.
Not all succulents are cold hardy, But there's a large group of Sempervivum, Sedums, and cacti that will all grow in freezing climates (down to Zone 4) that get several inches of snow in the winter.