The claim that FDR engineered a recovery is based on the fact that Gross Domestic Product (GDP) grew and unemployment dropped from 1933 to 1936. While these statements are correct they hide the fact that during that time GDP never reached 1929 levels and that unemployment stayed in the double digit range. The key question, though, is why did unemployment go up and GDP go down in 1937 and 1938?
The answer is that the economy never really recovered. The government's massive spending was not recovery, but as John Flynn put it "a palliative". By 1938,
The country had now really reached a greater crisis than in 1933. The public debt, which was 22 billion when Roosevelt took office almost all a heritage of World War IIwas now 37 billion. Taxes were more than doubled. The President had a war on against the conservatives in his party and his own cabinet was split and angry. Unemployment was several thousand more than it was in October, 1932. Roosevelt knew now he was in a crisis. And he had at his disposal nothing to fight it with save a weapon government spending which had failed and which he felt now was a palliative and not a cure.*All of FDR's spending had failed to revive savings and business investment, without which there can be no economic growth. (See here.) To use an analogy, a patient in a coma is on life support. Machines are forcing the pumping of the heart, the inhaling and exhaling of the lungs. While this may even result in better vital signs than the patient previously had, it is far from being a recovery. Under these circumstances it should come as no surprise that the patient dies if life support is removed.
So it was for the economy during the Great Depression. When government spending slowed and taxes rose the still comatose economy took a turn for the worse. The obvious lesson for us today is that we don't want to repeat the mistakes of that era and try to spend our way to prosperity.
* "The Roosevelt Myth", pg. 122