They're not even close to being the same. Functionally, wrought iron can be forged and cast iron would crumble if you tried it. It is suitable only for casting. Wrought iron gets it's name from being forged, or wrought, to reduce the impurities and carbon content in it. Wrought iron also welds much easier. Cast iron welds with a torch or a stick welder with difficulty and not at all by forge welding. It wouldn't surprise me at all if welding cast iron had it's own certification separate from steel welding. Chemically, both have silicate strands throughout them. Wrought iron has essentially no to up to 0.2% carbon, though I think wrought with any measurable carbon content is also referred to as wrought steel, reserving the term wrought iron for the product without any carbon in it. Cast iron has a carbon content of over 2% and may have graphite inclusions as well as silicate strands which makes it much more brittle. It's only slightly more refined than pig iron, which is iron as it comes from the furnace.
Doug