Cast and speed are both attempts at measuring performance. The term cast is well recorded in history and simply means how far an arrow is cast, thrown, kicked or flipped. Distance was the only measurable method of comparison before speed became measurable and "how far can you cast an arrow" was shortened to "cast."
Once the ability to measurable "speed" was added to the mix, it created two measurable factors, fps and measured distance. The human nature to "compare" stuff will always be part of our being and folks use one of the other to prove "their" bow is better.
Cast and speed are not the same but related. Cast is simply the distance an arrow is shot. Speed if how many fps the arrow was traveling at some distance, generally pretty close to the bow.
Cast cannot influence the bow, it simply measures the distance an arrow was cast by the bow. Speed, on the other hand, is a factor that can influence the cast or distance an arrow can be shot from a bow. There are other factors besides speed that can influence the cast/distance an arrow can be shot as well.