Econometrics and Statistics - What's the Difference?

Apparently, I'm not the only one who finds this hard to explain:

Because of the prevelance and power of the Law & Economics movement in legal scholarship, there was particular interest in the difference between statistics and economics/econometrics. I had a certain amount of trouble answering the question. It was easy to point out that the best quantitative empiricists move within all fields and are able to read all literatures. As an aspiring statistician, it was also easy to give the statistical version of things, which is that statisticians invent data analysis techniques and methods that, after ten to twenty-five to forty years, filter into or are reinvented by other fields (whenever I said this, I clarified that this story was a caricature).

So what is the difference between an empirical, data-centered economist and an applied statistician? The stereotypes I've internalized from hanging out in an East Coast statistics department are that economists tend to focus more on parameter estimation, asymptotics, unbiasedness, and paper-and-pencil solutions to problems (which can then be implemented via canned software like STATA), whereas applied statisticians are leaning more towards imputation and predictive inference, Bayesian thinking, and computational solutions to problems (which require programming in packages such as R). Anyone care to disabuse me of these notions?

(HT Marginal Revolution)