The arguments in favor of R boil down to this: R is more widely used by statisticians and has a much larger library of statistical packages. But R is not a very good programming language, is difficult to learn, and is not well suited to integrate with or be used for more general purpose programming tasks.
Python, on the other hand, has a vast library of packages but does not yet have nearly as many packages specialized for the statistical computing domain. The arguments in favor of Python are, in essence, that it's very easy to learn and easy to use and easy to integrate with other general purpose programming tasks. Python is also gaining a lot of momentum in the scientific computing community. For many statistical analysis applications (most?), the packages that do exist for Python are more than adequate. Some folks even suggest that R's lead over Python is evaporating fast.
As a domain-specific language, it's really great. The functional programming features have a great syntax (IMHO) for doing math stuff. As a general purpose language, it's awful. The library is large and easy to use, and the documentation is a pleasure to read. It's also as proprietary as it gets. Wolfram Research tries to ease the pain of that constricting noose with the CDF player and the ability to embed certain kinds of Mathematica--er, I mean, Wolfram Language code into web pages. In practice, however, everyone you want to share your code with is going to need to buy a Wolfram Research product or work at an institution that has a site license.
Soooo....all you do is GUis.
Do you even know what Qt is? It's way more than just a widget toolkit.
Sagemath is not just freeware but actual open source, and it is not even that, it's just a repackaging of existing software packages IIRC.
This is very incorrect. Sage's website accurately describes it: "It combines the power of many existing open-source packages into a common Python-based interface. Mission: Creating a viable free open source alternative to Magma, Maple, Mathematica and Matlab."
Again, the problem is NOT a problem of AUTHORSHIP. Authorship is easy. It's a problem of DISPLAY. And it is a serious and important problem to be solved. The web was invented to share scientific information. Education on the web is huge--and growing. Academic publishers, mathematical software, and software shims that display math in a browser all use MathML extensively. It's a ubiquitous technology precisely because it fills a need in the industry, and it fills it well. What's more, MathML is important for an accessible web.
PDF is clearly not good enough for digital consumption. PDF is great for print but totally sucks for screens. MathJax is amazing (as are the people behind it), but it is a huge, complicated, and inefficient solution to the problem of math in the browser. The author of the linked article in the submission works on MathJax professionally and is advocating MathML support in the browser. That should tell you something. (In fact, MathJax itself uses MathML both internally and as an input/output format.)