If you look at the dictionary meanings of the terms, you're correct: "Wissenschaft" means science in general, "Naturwissenschaft" means natural science. But often the more general term "Wissenschaft" is used to refer only to "hard science", meaning natural science.
This can probably be best understood if you look at the idea of the two cultures, that the British scientist and novelist C. P. Snow brought up in 1959. According to that idea, the intellectual life of the western societies was generally split into two cultures - the (natural) sciences on one hand, the humanities on the other hand. Snow described that dichotomy as a major hindrance to solving many problems.
For a long time, many people sort of looked down on humanities and didn't consider them "real sciences". After all, could social sciences, political sciences and so forth really prove their hypothesises like physics or chemistry could? Natural sciences were "hard science", verifiable, dependable. That "artsy stuff" that the people over at the humanities building were doing was mostly a matter of opinion and belief, maybe true, maybe not, wasn't it?
Over the decades, the gap got smaller. People learned that the insight in natural sciences has limits as well, and that the humanities has its own tool box of methods to underpin its claims. But, especially in media coverage, "Wissenschaft" is still used to refer to natural science only in many cases. As anecdotal evidence, I checked the first ten articles found on Google News for the search term "Wissenschaft" earlier today. Eight of them covered topics from natural science, but used more general terms like "Wissenschaft", "Wissenschaftler", "wissenschaftliche Studie" only. The remaining two were about the recent decisions in the Exzellenzinitiative which universities get additional money, that to my knowledge includes humanities as well.
(Because this question has been reopended again, I reworked my comments above into an answer.)