As the heading suggests: limestone caves and karst caves are definitely not the same. There is both, limestone caves which are not karst caves, and karst caves which are not in limestone.
So why do many people use this two terms synonymously? Because most of the caves on earth are limestone karst caves! This amount is so overwhelming, all other kinds of caves seem to be just strange exceptions of little interest. So most people think, karst caves and limestone caves are the same.
Why is this difference between the two terms? Because one describes the rock the cave is in, and the other one describes the mechanism how it is formed. There may be both, the same formation mechanism working in different rocks and different mechanisms of cave formation in the sam kind of rock.
Limestone caves are caves located in the rock called limestone. The term does not tell anything about the process how they are formed. Of course, limestone is soluble, and so most caves are karst caves formed by solution. But there are caves like tectonic caves, which are formed by rock movements, and which occur in all kinds of rocks, including limestone. Erosional caves are formed by mechanical erosion, a river or the wave, in any kind of rocks including limestone. There are also primary caves like tufa caves or reef caves in limestone, which are no karst caves.
Karst caves are formed by solution by water in any water-soluble rock. There are various rocks which are soluble, including ice, salt, gypsum (or anhydrite), limestones, and even quartzite and granite under certain circumstances. All those caves are called karst caves.
Limestone karst caves are formed inside several varieties of limestone. The rocks may be:
So why are there so much limestone karst caves? It is rather easy: limestone is a very common rock and the forming of karst in soluble rock is also very common. Limestone is a very common sedimentary rock, formed by the remains of all kinds of animals. The other soluble rocks need special climates for their forming, like arid climate. Limestone is produced nearly everywhere on the World: in tropic reefs consisting of limestone corals or sponges, in shelf areas and in large lakes. It is not formed in deep sea, as the deep sea water solutes the limestone falling down from the upper layers of the sea. But it is formed in the shallow seas which precedes orogenesis (the forming of mountains) and it gets lifted during the orogenetic process. After the exposure of the limestone it is subject to weathering which typically means the development of an Karst area.