First, let's state that obviously there will be those who'll say "what confusion" (by confusion I mean for example questions like "what philosophy has to do with science"), and for them I'll start with my own experience: as someone who studies philosophy independently, at first it seemed very odd to study philosophy to approach questions that seemed to be in the scientific realm, and I'm (very) often asked "why is [insert topic I talked about] philosophical and not scientific?" (and that's when the other person is accepting of listening to other approaches, most of the time it's more like "this ISN'T philosophy!"). And sure, you can say that I hang out with the wrong people that are simply ignorant of philosophy, but we can still see that many students are thinking the same way, at least at first (we can see it indirectly when we look at the number of majors in philosophy, as seen here, and for example in article such as this).
So now that we've shown that the issue surely exists, I'd like to propose, and ask if that can be said to be true, that the confusion comes with the rise of the analytic movement, and more precisely the logical positivism it has entailed (most of the time). (*Edit: thanks to Christopher, I'd like to emphasize that the entailment of logical positivism I mean here is mostly at the beginning of the movement, and not so much at the present day, as we know it's more complicated nowadays, but the public isn't much acknowledged about it, nor should we expect it to be.)
Let's start with the facts first: We know that philosophy has branched scientific departments over the years, notable examples from the last 200 years would be physics and psychology. We also know (I take it from a writer's note from Hugo Bergman's book "History of Modern Philosophy") that at least in the 18th-19th centuries (and probably even sooner), the public, especially the educated high society, took much interest in philosophical inquiries (the example Bergman gives is the Leibniz-Clarke Correspondence).
Without saying the interest in the big philosophical questions (God's existence, life's meaning, et cetera) has significantly (if at all) decline, we obviously see a look towards answers from the scientific departments rather than the philosophy department - which, as we've stated, is natural, as philosophy branched out to those more specific departments that could probably answer questions more appropriately. But, as stated in the first paragraph, we do see a lot of confusion about philosophy's place in the academia, so we may ask, why is that? If not from lack of interest in those questions, why does the the interest in philosophy decline?
One answer would be because, as we've stated, those questions seemingly moved to the branches we talked about. But I think most of the people here would agree that answering these questions from the limited scope of the branches is wrong, as we still need the appropriate philosophical work on the questions.
But let's take a step back and see why people consider the branches to be enough to answer the questions. For me, it seemed quite intuitive that the reason is the logical positivism that was being pushed by analytic philosophers such as Wittgenstein and Carnap (I'd include Russell too, or at least his legacy) - logical positivism (in a very short and relevant description) entails that we can trust the natural sciences to solve the philosophical problems we used to have.
I can also state that it seems as though in the continental legacy this issue never rises: continental philosophers keeps on mixing science and philosophy (or mostly, metaphilosophy) without really worrying about the connection between the two, as though they naturally complement each other. (yet, our Western Civilization, or at least the English-speaking countries, is mostly influenced by the analytic movement, so we see more effect on the public from the first kind rather than the latter.)