According to the United States Environmental Protection Agency, there are 12 principles of green chemistry. The second method of synthesis is more green than the first method of synthesis as it adheres to the first, second and fourth principles while the first method does not. We can easily observe why this is so.
Firstly, the second synthesis method prevents wastage of any reactants, all the reactants go into the making of the product, copper (I) iodide while for the first method, there are other products generated along with the desired product. In other words, there is a wastage of reactants if the first synthesis method is used.
Secondly, the second synthesis method maximises atom economy (i.e. waste few or no atoms). This is similar to the first principle. Atoms are wasted in the first synthesis method in making the other products.
Thirdly, the second synthesis method produces what is desired and does not produce any other chemicals that may be hazardous. According to this and this, sodium tetrathionate and sodium thiosulfate are irritants to the human body.
Hope I have answered your question.