Open science is the umbrella term of the movement to make scientific research, data and dissemination accessible to all levels of an inquiring society, amateur or professional. It encompasses practices such as publishing open research, campaigning for open access, encouraging scientists to practice open notebook science, and generally making it easier to publish and communicate scientific knowledge.
The term reproducible research...refers to the idea that the ultimate product of research is the paper along with the full computational environment used to produce the results in the paper such as the code, data, etc. necessary for reproduction of the results and building upon the research.
The two ideas aren't necessarily connected: putting research out in the open doesn't automatically make it reproducible, and someone can do the work required for 100% computational reproducibility without sharing it with the world. However, advocates of one often advocate the other as well, and they do seem like a natural pair.
Computational competence is like that too. It's possible to do reproducible research without knowing how to program, what version control is, etc., but those skills make it a lot easier. Similarly, it's possible to adopt open notebook practices without understanding what's going on behind the curtain, but only if everything works properly every time: as soon as something doesn't, you need to understand the HTTP request cycle, what an API is, or how to write simple database queries (or have a labmate who does). Everything Software Carpentry teaches can be used to support opaque science, but I believe that lowering the practical barriers to adopting open, reproducible practices will speed their wider adoption. And in my opinion, that's a good thing...
Later: in response to an emailed comment, yes, I do see an analogy with the invention of printing (and later, of the web). Gutenberg and others didn't set out to foment religious and political dissent, but by putting the means of mass communication in the hands of the masses, they made it easier for those with opinions to make those opinions known. Teaching researchers how to build things themselves doesn't necessarily mean the end of price gouging by big publishers, or more trustworthy computational science, but a world in which 50% of people can do something is going to be very different from one in which only 1% of them can do it.
Originally posted at Software Carpentry.