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 same 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 waves, 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.