There are people who use the words "problematize" and "problematic" (when used as a noun) as shibboleths to identify postmodernists. They aren't entirely wrong about this. Postmodernism does indeed circle about problems in a distinctive way. It rejects the idea that our job is to solve problems or answer questions; rather, it suggests, "It is enough that the question should be posed with sufficient force" (Deleuze, DR, p. 107).
Modernists search for solutions. They pose questions in order to answer them. Indeed, the "problem" is usually ignorance of some set of facts, and the solution, then, is simply to obtain knowledge of those facts. As I usually say, that's a perfectly respectable project. But it is not the only possible one.
One not quite postmodern alternative is Wittgenstein's approach to "philosophical problems". He believed that philosophical problems are different from other kinds of problems in that they don't have solutions; rather, what is needed is a dissolution of the problem. To free ourselves from it, we must show that the apparent problem is not really a problem. Philosophical problems are pseudo-problems, false problems, and the philosophers job is to expose their falsity.
Postmodernists do not abandon the search for solutions on the assumption that the problems aren't real. On the contrary, they acknowledge the reality of the problem more strongly than the reality of the solution. Drawing on the experience of "neuropaths and psychopaths", Deleuze puts it as follows:
They bear witness to that transcendence, and to the most extraordinary play of the true and the false which occurs not at the level of answers and solutions but at the level of the problems themselves, in the questions themselves—in other words, in conditions under which the false becomes the mode of exploration of the true, the very space of its essential disguises or its fundamental displacement: the pseudos here becomes the pathos of the True. (DR, p. 106)
This somewhat romantic attitude to mental illness is no longer fashionable, but it is still an essential part of the postmodern attitude at one level. Even madness should not, says the postmodernist, simply be construed as a "problem" to which psychiatry is then entrusted to find a "solution". Rather, madness should be explored as a "problematic" in its own right.
Simplifying somewhat, we might say that in Madness and Civilization, Foucault turned the problem of psychiatry on its head. He sought the problem to which madness is a solution. He did not want to solve the problem of insanity; he wanted to "problematize" the psychiatric solution. To problematize is simply to elaborate the "essential disguises" of the true, to trace its "fundamental displacement". To solve a problem, by contrast, is to expose the truth, to nail it down.
As always, the difference here is one of attitude, a difference of mental style. It is as old as the difference between "classical" and "romantic" writing. I, for one, do not believe that the question of whether we are postmodern or modern (or what we should be) can be settled once and for all. Sometimes we solve this problem in our writing and sometimes it remains (a) problematic. Both require strength in prose to face.