What makes the difference between wrought iron and soft steel?
History and technology advancement
The modern equivalent of wrought iron is mild steel.
Language barrier is frustrating!
'Wrought' means worked... its roots are Germanic! To this end... I feel like 'wrought' iron can be applied in many situations - for instance, hammered mild steel is in some senses 'wrought'. It is also a style - 'wrought' iron decorations are very popular in victorian-era fencing (often with cast-iron filigree balcony decorations). The old material would have been wrought iron... I am sure the new replicas are mild steel.
Wrought iron was made by decarburising high-carbon content alloys (e.g. cast or pig iron) through oxidation. Indeed, as you say, one of the early mass manufacturing methods was through a
puddling furnace (the 'Puddeleisen'). Prior to that, bloomeries produced a range of iron alloys across the carbon spectrum. A blacksmith could choose a part of the bloom that suited their needs... or even combine high and low carbon bits of the bloom. Common to these methods is that the process was not particularly clean. The products were not homogeneous and would contain slag inclusions.
To homogenise the steel and remove the slag, the metal was worked (wrought). Because the steel is 'dirty' by modern standards, you can see impurities from the smelting process. By working the steel, the resulting stock can have a nice grain and a layered, pattern-welded look from the folding/working. The contaminants can be really subtle (my preference)... or outright gaudy (meh)!
I doubt wrought iron is manufactured in commercial quantities. Industry would prefer 'clean' mild steel. That is why blacksmiths might collect old steel from railroads, farm equipment... old bridges or ships. The history behind the material can be a nice touch. Seriously enthusiastic blacksmiths might make their own wrought iron from DIY bloom steel.