Well, in the spirit of rehabilitating the discussion, I note that, in analytically oriented seminars led by leading analytic philosophers, I have studied
Meinong, Husserl and Brentano. My own work on dreaming explicitly quotes Sartre, as does Colin McGinn's recent book on imagination.
Phil mentions John Searle's book Intentionality, which depends heavily and explicitly on Heidegger. I haven't much grip on contemporary
French philosophy, but I certainly have nothing against the Continental philosophy I have read. I think the phenomenological
stuff is especially interesting. I'm not acquainted with a
deep prejudice by analytic philosophers against the Continental, though there used to be one back in the days when
Positivism was ascendant. Doubtless it happens but I think the serious study of Continental philosophers by analytic philosophers
speaks more accurately to the analytic side's estimate of Continental philosophy.

Let me say something about the analytic tradition. It begins IMO, with Frege's logic. If Aristotle was logic's Galileo, Frege was its Newton.
This extraordinary development, especially coupled with Russells use of it to apparently solve some philosophical problems,
led to serious concentration on language. The new logic, some thought, would solve philosophical problems by clarifying
language. This led to the idea that philosophical problems were entirely confusions of language. In addition, some thought
that the reason Frege's logic was so powerful was that its structure corresponded to the structure of reality, so that
we could read off metaphysics from the new logic. In the midst of this heady stuff, the positivists coupled Frege's logic
with Hume's empiricism, we get the Verifiability Criterion of Meaning, the rejection of metaphysics, the rejection
of ethics, the rejection of most of the history of philosophy (metaphysics), the rejection of the Continental.

The Positivist's  program begins to lose force in the 50s. The logical and linguistic skills developed
are used to re-introduce metaphysical questions (especially Kripke and the interest in modal logic). It is felt we can
now make progress on old
issues because we have the logical means to do it. Metaphysics comes back with a vengeance, modal questions,
personal identity, free will, God's existence, universals/nominalism, propositions, parts-wholes,
the nature of time, the question of persistence through time,
and so on. History of philosophy returns with extraordinary vigor, scholars trying to understand
the great philosophers in reference to their concerns in their own day. The Vietnam generation
of philosophers rejects entirely the positivistic view that philosophy shouldn't comment on moral
issues, and there is an explosion of applied ethics and medical ethics, which I detailed earlier in this thread.
Philosophy of religion also comes to the fore. Further analytic philosophers are surfing off science, especially
developments in neurology and computer science/artificial intelligence, which make it possible
to address the old mind-body problem in a more informed way. Also the metaphysical debate
about the nature of time (which goes back to Augustine) is affected by physics and relativity theory.  Political philosophy also
advances, especially through John Rawls' work. There is a mini-industry of
philosophers working on the meaning of life. And epistemology is very important,
as is probability theory and the philosophy of science. Philosophy of biology....
And more...

Amid all this diversity, running like a thread through it, is Frege's logic. Sooner or later analytic philosophers
trying to clarify their thinking, will see how it looks in first-order predicate calculus. Also the discussion
of language and logic, plus set theory and mereology, have provided means to express difficult issues much
more clearly. So what one has is the continuation of the traditional philosophical enterprise, mostly old problems
pursued across a wide front, with an arsenal of new logical tools, not always at the fore, but
underlying the enterprise. If Aristotle came back today, or Aquinas or Hume or Descartes or Leibniz or Anselm or....
I believe they would join in with gusto.

The analytic tradition is precisely the sort of thing that happens after a major advance in logic, which
raises issues of its own and also enables us to see more deeply and clearly into old issues.

The analytic tradition involves a passion for clarity, that arguments will be set out clearly so that fallacies, if present will
be discovered.   While there is sometimes considerable craftsmanship and elegance in writing, there is
little rhetoric.

I think it's fine that people pursue philosophy in other ways than this,
and I have noted above the real interest in Continental philosophers among
analytic people with whom I've studied. It is widely recognized that some
Continental philosophers were grappling intelligently with real problems
and that we can learn from their efforts.