Carbon steel and cast iron skillets are fundamentally different instruments meant for different uses. While they can look almost identical, they serve separate needs.
Carbon steel skillets are made from a combination of iron and carbon, this makes them lighter and more responsive to temperature changes than cast iron skillets. They are also more durable and less prone to cracking than cast iron skillets. Carbon steel skillets are ideal for searing, sautéing, and stir-frying, as they heat up quickly and evenly.
Since cast iron skillets are made by a different process, they are poured into a mold, they heat up more slowly and retain more heat. This makes them a great choice for slow cooking and certain baking recipes. Food can overcook in cast iron because the pan retains so much and it's important to adjust recipes to account for it. Basically the choice of cast iron or carbon steel depends on what you're cooking. If you want a skillet that heats up quickly and is ideal for searing and sautéing, a carbon steel skillet is the obvious choice - think stir fries, seared ribeye etc.
If you prefer a skillet that retains heat for a longer time and is great for slow-cooking and baking, grab your cast iron skillet and get to work.