It seems that they are very close to one another in their methods, and their beginnings. First off, science began as natural philosophy, so historically they are one in the same. Furthermore, the scientific method is just a refined Socratic method, in that it is used to deal with nature, rather than the more abstract things that philosophy deals with (ethics, metaphysics, religion, etc). However, even though the questions of philosophy (at an academic level) is more relevant than that of the questions of the sciences (at an academic level), as philosophy deals with moral decision making, while science deals with the physical properties a black hole has, the latter is more popular than the former. I don't mean to discount science, but I am interested in the history between the two, and how science ended up on top. So, my question is, why did science become more popular than philosophy?
There's two assumptions you're making that I don't understand:
How are you judging popularity? Are we talking number of faculty/students in the fields? Then yes, the physical sciences are much more popular. But this is a poor comparison and hardly a point worth making. We have a Philosophy department, but we don't have a "Physical Science" department. Since philosophy is so conceptual and theoretical, this is possible. The work done in a philosophy department can be extended to numerous applications since it deals with general principles. The general principles of science, however, are well established and not frequently subject to review. The work done in a Robotics Lab will largely not be applicable to a Botanist. Ethical principles from a philosopher, however, might affect both in a similar way. We need scientific specialists; "philosophical specialists" go by other names which leads into my next point...
You're creating a dichotomy I don't believe exists. As you note, philosophy informs all fields. It manifests as the scientific method, the Hippocratic oath, and as NAGPRA guideline for proper treatment of archaeologically excavated human remains. Without philosophy, the social sciences have no theoretical ground; you can see my write-up on history and philosophy here. Pure philosophy must be distilled into practical relevance. As said above, we may better know these specialized departments as Public Policy, Law, Political Science, Religious Studies, Seminaries, Medical Ethics, Education Studies, and the like. Just because they don't have Philosophy in their name doesn't mean they aren't heavily engaged with, dependent upon, and contributors to philosophy. Even as a scientific minded anthropologist, I need to know my Marx and Bourdieu since they provide so much of the theoretical framework. Beyond basics like the scientific method, this level of integration with other fields simply doesn't happen with most sciences.
TL;DR: Pure philosophy can exist since it is more fluid and worth debating than the nonexistent field of pure theoretical science. Science has a broader range of independent fields, whereas philosophy is integrated and integral to a variety of applied fields. They cannot be compared as two equal concepts.