Reindeer and caribou belong to the same species, which is Rangifer tarandus. In general, people use “caribou” in North America, “reindeer” in Europe and Asia.
“Caribou” also tends to refer to the larger, wild R. tarandus types, like the ones in Canada and Alaska, while “reindeer” often, but not always, means the slightly smaller, domesticated kinds, like many of the ones in Scandinavia.
But a 2013 study complicates matters. Based on DNA analysis, the study suggested that reindeer/caribou should be in two groups. One group is the caribou in southern Canada. The other group is the caribou in Alaska and northern Canada plus the reindeer in Europe and Asia.
The scientists who did the study said the Ice Age split the two groups apart about 200,000 years ago, and their genes and adaptations to their environment, including to changing climates, are somewhat different because of it.