To put it somewhat simplistically and anachronistically, Wittgenstein was an original post-modernist lurking at modernist Cambridge. Popper was something of a rationalist holdout. Of course Popper's political philosophy, because it so boldly attacks Marx, does not stand him in good stead with much of the left. Ayn Rand he was not, however.
Wittgenstein seemed to understand the 'continental philosophers' (e.g., Bretano, Meinong, Husserl) that the Anglo-analytic tradition has so often dismissed (going back at least as far as the anti-Hegelian phase).
However, if you want to understand the post-modern philosophy of, say, the post-structuralists, a strong grounding in these continental philosophers is necessary.
As for Popper's philosophy of science? Epistemologically speaking, it's now thoroughly post-modern. Popper found Kuhn far too radically skeptical, never mind Feyerabend (F's perfect contemporary is Baudrillard). It's always interesting as well that so many who take Anglo-analytic philosophy to new places are not very Anglo (Peano, Frege, Wittgenstein, Godel, Popper, Tarski, Feyerabend).
Still, contrary to popular belief in North American 'English' departments, post-modern analytic philosophy and post-structuralism have far more in common than is often supposed.
As for Wittgenstein's problems with depression, this type of problem seems to go with creative personalities.
BTW, Derek Jarman's film on Wittgenstein is far better than I was led to believe from reading interviews with Terry Eagleton on it (who was supposed to be some sort of adviser who later dissociated himself from the project). Just wish someone hadn't stolen my only copy.
Charles Jannuzi