Horse chestnuts and chestnuts are different genera, though both have "American" and "European" species. The American horse chestnut is often called the "buckeye". The nuts are vaguely reminiscent of each other, but the horse chestnuts and real chestnuts aren't closely related.

This may seem pedantic, but it's sort of important, as real chestnuts are edible (and delicious), while horse chestnuts are toxic to human beings.