Is computer science pure science or just applied science?
My view is that what is and isn't "science", pure or applied, is determined by method.
Pure science is characterized, broadly, by the scientific method. Different branches take it to different extents and in different directions, but they all share a similar core: theories based on observations of the world and tested empirically and quantitatively either through experiments or, at least, observational studies.
Pure science answers "what" by observing the real world.
Applied science, on the other hand, is closer to engineering. It's about coming up with specific ways to accomplish some goal. The goal is to take a scientific understanding of the world and use it to develop solutions to problems.
Applied science answers "how" by inventing based on scientific principles.
So where does this leave computer science?
Well, CS is an interesting field. It isn't defined by how it approaches problems, but by what topics it covers—anything remotely related to computers will do. So it's both at once, and more. CS sub-fields lean in all sorts of directions: pure science, engineering or even abstract math. Sometimes they even veer into art!
CS encompasses both pure and applied sciences to do with computers and computation, and more.
In fact, you can even see a surprising range even in a single sub-field. The study of programming languages is a great example. Practical PL design and tooling is an engineering discipline at heart—we invent languages and tools based on a broader understanding of the subject. PL theory, on the other hand, is math—as pure as it comes. (In fact, it turns out that PL theory can provide a coherent foundation for all of mathematics!) And a small, emerging amount of work is scientific in the same way as psychology, trying to build up an experimental understanding of how humans learn languages and using that to inform language design.
Looking in other sub-fields, you'll find other mixes of approaches and philosophies. But, taken as a whole, CS thinking is surprisingly diverse, largely because it's defined purely by its subject, not its method.