Knowing that a reindeer and a caribou are the same thing might seem like a small thing, but it is important to educate children properly and cover all topics. There is no reason not to teach it, even though most North Americans refer to caribou as reindeer, particularly around Christmas time. It seems like something that should be taught to provide information to children.
Some schools do teach that Caribou is the North American name for reindeer, but when children are young it can be hard for them to learn multiple names for the same thing. As they get older though, they are often taught that caribou and reindeer are differnt words for the same thing.
Why mess up little kids minds by telling them that reindeer are actually just caribou by another name? Eventually they will come across it in books or on TV, and will understand that it is the same animal. When they get older they can learn about it in college or by reading more.
While all herds and subspecies of reindeer in North America are caribou, the word itself is not a "North American" name. The name has French roots. It shouldn't be taught that the word itself is derived from North America, because it isn't; however, all species and herds of reindeer in North America are caribou.