Simon Wren-Lewis has a good description of Helicopter money but I think he may be missing a couple things.
First is the perception of a "tax cut" vs. "free money". At one level, if everyone gets $1000 it is the same either way. If it is a tax cut, however, there is a perception that people who pay more in taxes are getting less out of the deal, and also, people who have little to no tax liability may receive nothing, and those are often the people most in need and most likely to spend. Additionally, people seem to act differently about free money (read: they spend it) compared to earned money (which a tax cut is returning).
The second thing is the targeting/distribution aspect. Tax cuts get money to all tax payers/filers. Helicopter money can get money to everyone. It could also target specific [debt] problems that may be holding back the economy (student loans, credit cards, mortgages). True the latter is not exactly the same as helicopter money, but it isn't the same as the current QE strategy either, and everything else he said about it applies equally.
Personally the way I would like to see "helicopter money" distributed is via debt reduction to real people. I am particularly fond of the fed buying and torching student loan debt, but I could also get behind credit card debt (which I don't really have) or mortgages, though with that last one I think I'd prefer a partial rather than complete buyout (up to maybe $50k per primary mortgage).
There is a lot of fear and uncertainty remaining in the US, and having lots of debt is a real issue for it. People don't feel as free to spend money if they have these huge responsibilities hanging over them. Even a fairly generous helicopter drop (via tax cut) isn't directly addressing the debt issue, and probably won't do enough to boost the economy.