While the overall composition is calcium carbonate, the specific form is typically calcite, which is a crystalline form of calcium carbonate.
It's important to note that the formation of dripstone isn't just about the type of limestone. It also depends on the presence of water containing dissolved carbon dioxide, which creates a weak carbonic acid that can dissolve the limestone. This dissolved calcium carbonate is then deposited over time as the water evaporates, forming the stalactites and stalagmites.