Damascus steel and carbon steel are not two different metals. Most Damascus is carbon steel: two carbon knife steels forge-welded together in many layers and etched so the pattern shows. So comparing "Damascus vs carbon" is usually comparing layered, patterned carbon steel against plain, single-steel carbon. The blade performs like the steels it is made from. The Damascus part is the pattern. The comparison that actually changes how a knife behaves is carbon versus stainless, and that is about corrosion, not pattern.
Here is how the terms fit together, so you can choose by what matters.
Is Damascus steel carbon steel?
Usually, yes. DSC® Carbon is two carbon knife steels, 1.2842 and 1.2767, forge-welded into 160 layers. Both are standard, modern carbon steels. The pattern you see is those layers, revealed when the finished bar is etched. Nothing is added that makes it a different kind of metal. A carbon Damascus blade behaves like a good carbon blade: it takes a very keen edge, it reacts with acids and develops a patina, and it needs drying and a little care.
The one thing "Damascus" does not tell you is the alloy family, because Damascus can also be stainless. DSC® Inox is a stainless Damascus, 1.4034 and 19C27 in 120 layers. So "Damascus" describes how the steel is built, layered and pattern-welded, not what it is made of.
What Damascus adds over plain carbon steel
The honest answer is: the pattern, and the work that goes into the forge weld. A Damascus billet is not automatically sharper, harder, or stronger than a single carbon steel. Those properties come from the steels in the billet and the heat treatment, not from the layering. A well-made mono carbon blade and a carbon Damascus blade of similar steels, heat treated the same way, cut much alike. For more on that, see Is Damascus steel stronger than regular steel?
What you get for the extra work is a blade with figure that runs right through the steel, no two pieces the same, in whichever pattern you choose.
Where the two are the same
Edge, hardness, and care all come from the component steel. DSC® Carbon reaches up to 62 HRC and takes a clean etch, the way its component steels do on their own. It patinas and can spot with moisture, the way any carbon knife steel does. If you have worked plain carbon knife steel, carbon Damascus will feel familiar at the bench and in the kitchen.
The comparison that actually matters: carbon vs stainless
If you are weighing your options, the useful fork is not Damascus versus carbon. It is carbon versus stainless, because that decides how much the knife will fuss over moisture:
DSC® Carbon (1.2842 + 1.2767) is reactive. It takes a fine edge and a living patina, and it needs drying and the occasional oil.
DSC® Inox (1.4034 + 19C27) is stainless. It resists corrosion, is approved for direct contact with food, and stays bright with ordinary care, while still hardening to a keen 60 to 61 HRC.
Both are Damascus. Both carry any of the patterns. The choice between them is about use and upkeep, not looks. There is a fuller walkthrough in carbon vs stainless Damascus and in Is Damascus steel food safe?
The bottom line
Damascus is how the steel is built, not a different metal from carbon steel. Most Damascus is carbon steel, layered and etched for the pattern, and it performs like the carbon steels inside it. Choose carbon or stainless by how the knife will be used, and choose the pattern by what you want to look at. Start with DSC® Carbon or DSC® Inox, cut to the inch.