I'm a huge carbon steel pan fan, since using them professionally for decades.
I got a nice Lodge cast iron pan a while back that I also love, but that I use differently.
For stovetop searing and finishing in an oven carbon steel wins out hands down, the pans get hotter faster and cool down faster for better temperature control, and are easier to throw around and take in and out of the oven. So I use mine for all quick searing jobs and also sometimes for searing off meats before transferring them to other recipients for longer cooks.
because of the better heat retention, and the usually thicker and higher straight sides of most cast iron designs, I find them better for longer sautéing and browning jobs like potatoes, or Spanish omelettes, or anything where you need a good sear and then to consume cooking at lower even temps. The seasoning on cast iron holds up better to these kinds of cooks as well, as simmering anything in a carbon steel pan will strip the seasoning quickly, if there is any acidic present in the liquid. Cast iron iOS more forgiving and the seasoning is generally more durable and thicker all else being equal.
For most home cooks I would say cast iron is more versatile and will get more use and be more useful if you want to have fewer pans in your collection. If you prefer to have more utensils and use the precisely best one for each occasion, I would say a carbon steel pan is a must have for it's incredible searing properties and precision.
I learned to cook with high stovetop heats and control my heat by taking the pan on and off the heat, I do this without thinking and it just works for me. A cast iron pan will destroy my wrists cooking this way.
Last point - cast iron wins hands down for cooking over campfires or a fire pit, I'll throw my Lodge on to my Weber to cook breakfast for a few people or to do some sides at the end of a BBQ cook, but I would never put my De Buyers over coals or wood, not out of fear of ruining them, but because cast iron is way better for the job.
I use De Buyer because I'm used to them and because they are cheap here, you have great carbon steel options that are US made, I'd compare those to De Buyer and buy what is less expensive near you.
De Buyer's Mineral B range have silicone coated carbon steel handles that stays cooler and fit home stovetops and smaller ovens easily, and have a coat of beeswax that keeps them from rusting before you use them the fist time (you burn this off when you first season them).
The Carbone Plus are the pro models found in commercial kitchens all over France and elsewhere in Europe, if you have a big range or a pro piano these are the bomb.
