[lutusp]

Such a formal comparison may exist somewhere, but I can summarize the key differences between Mathematica and Sage, two tools I'm familiar with.

Sage relies on Python, Mathematica relies on a proprietary language.

Mathematica can do a number of things that Sage can't do, but Sage has a wider array of mathematical tools and environments bundled with it (and Sage is a big package -- 4.4 GB installed). Sage includes a number of specialized packages for research and scientific mathematics beyond the most often used functions and packages.

Sage can perform the usual symbolic math operations -- solve equations, derive, integrate, produce many kinds of symbolic and numerical results. Mathematica can solve more cases in the same domains.

Sage is free. Mathematica presently costs $2,745 for the default, single-seat license.

Sage: http://www.sagemath.org/

My Sage tutorial: http://www.arachnoid.com/sage

Mathematica: http://www.wolfram.com/mathematica/pricing/industry-individu...