One of the reasons I chose to study philosophy at the University of Chicago was that institution’s reputation of being strong in both the analytic and the continental traditions. However, for that reason to be at all understandable–let alone convincing–to a non-philosopher I needed to be able to distinguish between the two without appeal to 
the two school’s major players. However, that is exactly how the two traditions are most often differentiated.
1) Analytic philosophy was founded by Frege, Russell, Moore, and Wittgenstein. Contemporary exemplars include the work of Carnap, Quine, and Kripke.
2) Continental philosophy was founded by Hegel, Heidegger, and Nietzche. Contemporary exemplars include the work of Gadamer, Foucault, Derrida, and Marion.
Again, unless you are already a PhD candidate in philosophy the preceding laundry lists are pretty much useless. Now, if you’ve studied a bit of philosophy in undergrad you might be able to recognize some names and perhaps figure out which camp your college or university’s philosophy department preferred–but that doesn’t help to explain to Aunt so and so why you are moving to Chicago instead of staying in Milwaukee. So blundering further, one could argue that:
1) Analytic philosophy’s grounds for authority tend to favor mathematics, logic, and the natural sciences (especially physics). Continental’s core concerns include what there is and how it can be known.
2) Continental philosophy’s grounds for authority tend to favor art, narrative, and certain interpretive methods. Continental’s core concerns tend to circle around exploring what it is to experience life as a human.
These distinctions get one closer to an answer that can be recited and understood in the grocery check out lane, but there is still quite a bit of overlap. So the question becomes, is it possible to further generalize and absolutize the difference between the two so that even Fox News could make the distinction?
1) A sample from Hilary Putnam’s classic analytic essay Brains in Vats.
Here is a science fiction possibility discussed by philosophers: imagine that a human being (you can imagine this to be yourself) has been subjected to an operation by an evil scientist. The person’s brain (your brain) has been removed from the body and placed in a vat of nutrients which keeps the brain alive. The nerve endings have been connected to a super-scientific computer which causes the person whose brain it is to have the illusion that everything is perfectly normal. There seem to be people, objects, the sky, etc.; but really, all the person (you) is experiencing is the result of electronic impulses travelling from the computer to the nerve endings.
2) A sample from Kierkegaard’s classic continental book Fear and Trembling:
The story about Abraham is remarkable in that it is always glorious no matter how poorly it is understood, but here again it is a matter of whether or not we are willing to work and be burdened. But we are unwilling to work, and yet we want to understand the story. We glorify Abraham* but how? We recite the whole story in cliches: “The great thing was that he loved God in such a way that he was willing to offer him the best.’* This is very true, but “the best” is a vague term. Mentally and orally we homologize Isaac and the best, and the contemplator can very well smoke his pipe while cogitating, and the listener may very well stretch out his legs comfortably. If that rich young man whom Jesus met along the way had sold all his possessions and given the money to the poor, we would praise him as we praise every great deed, even if we could not understand him without working, but he still would not become an Abraham, even though he sacrificed the best. What is omitted from Abraham’s story is the anxiety, because to money I have no ethical obligation, but to the son the father has the highest and holiest.
So, within the analytic tradition one creates fictions that imagine the world to be a way it is not and then considers what the similarities and differences between our world and that world might tell us. Within the continental tradition one tells a story about the world and then considers what the similarities and differences between the character’s actions and our actions in the same situation might tell us. You say imagine, I tell a story… you appeal to a hypothetical premise, I appeal to a narrative. Can’t we all get along?
(My most sincere thanks to Neal De Roo for pointing this out long before I could really appreciate how right he was!)
maphmaticallyyours 
Boston College is hosting a “contemporary philosophy seminar” this weekend on the subject of the Continental/Analytic Divide. So we’ve all been thinking and arguing about it for the past few days.
From my perspective (if I can use your post as an excuse to hatch some thoughts I’ve been incubating) as I’ve slowly gained access to the riches of, for example, analytic philosophy of language, post-analytic philosophy, remarkable contemporary “analytics” like Robert Brandom and John McDowell, and superb analytic historians of philosophy like Frederick Beiser, I’ve begun to see that with respect to still-pressing contemporary problematics (especially language and mind) one can without much trouble at all draw heavily on work on both sides of the divide. But in the problem of “facticity and validity” — as Habermas puts it — the primarily social problem of actual human life on the one hand and our desire for rational organization / justification on the other, the divide has horribly mangled a problem, such that one can only with difficulty work on it from just one side or other of the divide.
But to put all my cards on the table, I’m beginning to suspect that The Divide is really only an intractable problem with regard to a very specific set of philosophers on both sides — the early positivists and those of their heirs who held on to the abjuration of metaphysics (most did not), and, on the other hand, post-Husserlian (and especially French) phenomenology, and then the late 20th century Nietzscheans. These two fairly limited sets (yet vastly influential, to the degree that a contemporary philosopher will tend to find his sympathies always lying more with one set than the other) almost seem the only really immovable, unbridgeable gap. For the rest, I’d like to say, it’s all philosophy and I aspire to equal access to both sides for conceptual appropriation.
I think that analytic philosophy has much to offer in as much as it is willing to disentangle itself from Wittgenstein’s claim that all ethical, aesthetic, mathematical, philosophical, theological, and causal judgments are meaningless. Now fortunately–as I think you would agree–few contemporary analytic philosophers still work in the tightly defined world that Wittgenstenian philosophy carved out for itself. If nothing else, the nineteen-sixty’s civil rights/ war protest culture could not allow analytic philosophy to hold up in its natural sciences styled bunker. Yet, I would agree with you that the question comes down to seeking out social facticity v. rational, positivistic proofs. The two traditions still approach the project of philosophy with an eye toward different goals.
I am actually rather enjoying both. I didn’t get into philosophy because I was a frustrated mathematician or physicist (Frege or Wittgenstein)–in fact I got into philosophy because I don’t enjoy algebraic puzzles. But, I have enjoyed reading Rawls’ Theory of Justice and the metaphysics of David Lewis and Saul Kriptke–all of which seem to take a rational/ positivist/ natural sciences approach to their topics.
The interesting thing to note is that if Philosophy is generally hated by the rest of the humanities–and there is strong evidence to suggest that it is–then the philosophy that the humanities hate is the analytic philosophy with its mathematical/ legal flavor. Does that make analytic philosophy too much of a science? or does it mean that the rest of the humanities just abhor rigorous, critical argument?