I assume you mean both of these words in the sense of "to throw." In everyday, modern US English, "throw" is the general word. It is the normal word. Use "throw" unless you have a reason to use something different. "Pitch" means "to throw accurately," and is usually used in connection with baseball. The pitcher is the player who throws the ball (at speeds of up to 160 kph) toward the batter, who tries to hit it. You will occasionally hear other uses: "What should I do with this?" "Pitch it," meaning "throw it in the wastebasket." "Cast" means "to throw something that's light in weight," sometimes "to scatter." By far the most common uses refer to fishing: a fisherman casts a line or casts a net. It's a somewhat old-fashioned word. The 1611 King James translation of the Bible refers to "a time to cast away stones," while a modern translation says "a time to throw away stones." It appears as part of the word "broadcast" (which has an old literal meaning of "scattering seeds").

In the sense of "to throw," right? You cast a fishing net, or an anchor; you pitch a baseball or cricket ball. Casts are less specific about exactly where they land, and my own sense is they are more often thrown overhand. A pitch is directed at a particular target and is often thrown with high power and speed. "I'm casting out everything to do with my ex-boyfriend, so I pitched it all into the trash."