I agree with pohnpei's post above and would like to add that another very important success of the New Deal was that it restored a measure of popular faith in the government during a time of severe economic Depression. If this had not happened, or there had not been a New Deal, political revolution, whether violent or non-violent, was quite possible. People who are economically desperate have nothing to lose, and are willing to consider political alternatives. This explains the popularity of Huey Long's "Share Our Wealth" program and the growth of the US Communist Party during the early 1930s.
A failure of the New Deal, in addition to not curing America of her economic ills, was that it relied almost completely on deficit financing - borrowed money - to pay for its programs. This began a cycle of national debt, made worse by World War II, that we have never recovered from, and it is now a serious national issue.
The major success of the New Deal, in most historians' eyes, is that it helped ease the Great Depression. Please note that I did not say that it ended the Depression -- only the start of WWII managed that. But historians tend to say that FDR's actions gave people hope and helped pull the economy up from the worst of the Depression.
You can also argue that the New Deal had lasting good effects. For example, it helped lay the foundations for a booming post-war economy by building infrastructure like the TVA. And it helped make people feel more secure through programs like Social Security.
The major failing of the New Deal was that it did not end the Depression. I think that there are other things you can say to argue that it was bad (it hugely inflated the size of the government, it got the government to intervene more in the economy) but these are not failures of the program. Those are bad side effects. I'm defining failure as when something does not achieve what it set out to achieve.