I mean, I know the difference in the properties.. but how are they produced? what is the difference in the method of production?
Very little similarity except they are both mostly iron. Pig iron is from an archaic process of steel making ; The product of the blast furnace, very high carbon and silicon. Poor physical properties ,about the only direct use would be weights. Wrought iron is essentially "pure" iron, soft,ductile . Not much direct use ,it can be drawn into wire which hardens and gives it some strength. However, many products today are called wrought iron , they are actually (low) carbon steels . Today very little actual wrought iron is made ,I expect the furnace ( eg. oxygen converter )just blows oxygen ( and Argon ?) until carbon is down to 0.02 %+/-, adding no silicon or manganese. A pilot plant size operation.