Why was there disagreement between Republicans and Democrats as to how to solve the USA's economic difficulties between 1929 and 1941?
There was disagreement because these two political parties have fundamental differences with regard to the proper role of government in the economy.
Republicans tend to be much more of believers in laissez-faire capitalism. They feel that an economy does best when the government leaves it alone. By contrast, Democrats are much more likely to prefer that the government intervene in the economy. They do not believe that the market can fix itself. Therefore, Democrats wanted the New Deal and perhaps more while Republicans would have preferred to have the government stay out of things and let the economy heal itself. They would argue the recovery would have been faster if this had been done.