What is the difference between empathy and sympathy?

For the most part, these two nouns are not used interchangeably, but often we encounter them in contexts where their nuance is diminished or perhaps not relevant, providing no obvious indication why one was chosen over the other:

In order to succeed, humanitarian efforts require a “Goldilocks” solution–just the right mix of force and charity, sympathy and structure, blind will and determined follow-up.
—Wilfred M. McClay, The Wilson Quarterly, Summer 2008

Given his rich familiarity with things European, it is not surprising that Mr. Lewis writes with sympathy and perceptiveness about Edith Wharton.
—Frank Kermode, The New York Times Book Review, 11 July 1993

He’s the good man here, as he was in “Good Morning, Vietnam” and “Dead Poets Society,” and he does a fine job of it: he shows the warmth and reticence and empathy that Dr. Sayer needs.
—Pauline Kael, The New Yorker, 11 Feb. 1991

But crying as an embodiment of empathy is, I maintain, unique to humans and has played an essential role in human evolution and the development of human cultures.
—Michael Trimble, The New York Times, 11 Nov. 2012

The difference in meaning is usually explained with some variation of the following: sympathy is when you share the feelings of another; empathy is when you understand the feelings of another but do not necessarily share them.