A Fossil Cave is a cave which has no cave river because it is higher than the karst water table.
Karst has caves, and those holes in the rock are interconnected and filled with water, which is called an aquifer. Above the water table of this aquifer, the caves do not have water, they are called dry or fossil, the scientific term is vadose cave. Below the water table they are completely filled by water and are called underwater cave, the scientific term is phreatic cave. Then there is the area around the water table, caves in this area are typically filled with air but have a river flowing through the cave. Those caves are called river caves, the scientific term is epiphreatic cave.
The common names of such caves are misleading, which is due to the fact that the name describes an aspect of the cave, but the term is normally used in another way, which causes misinterpretations. The term Fossil Cave describes that cave formation, the enlargement of the passages, has stopped, and so the cave now fills with speleothems. It has nothing to do with the appearance of fossils, and the cave is also not "dead" or unchangeable, as the term is sometimes misinterpreted. The term Dry Cave describes the cave has no river and is not flooded, nevertheless, it is not dry, there is dripping water and a high humidity. The water from the surface must cross this zone on its way down to the groundwater. The best term to name this type of cave is obviously Vadose Cave, because it is located in the vadose zone, which is the upper part of the karstified limestone above the water table. Unfortunately, it is rather uncommon as most people have never heard that term.
Cave formation has four phases, solution of limestone underwater, erosion by a cave river, the cave river vanishes and speleothems form, and finally the cave reaches the surface, the ceiling collapses, and in the end the cave is gone. The last two phases are vadose caves. Obviously, vadose caves are comparably easy to explore and transform into show caves. There is no need for diving, no swimming, no floods which are dangerous and destroy trails and light. In other words, the overwhelming majority of show cave is of this type. However, cave systems often have all three cave types, and the show cave may offer a glimpse into a lower, water filled level.