Computer algebra systems, such as Mathematica, Maple, SageMath, Matlab's Symbolic Math Toolbox, etc. are very useful for the following tasks (ranked in importance):

• 1 = Deriving and manipulating exact closed-form quantities or special functions, including pattern recognition
• 2 = Numerical scratchpad for exploration
• 2 = High precision computations and variable precision arithmetic
• 3 = (Floating-point) Numerical analysis and numerical solutions

In terms of alternatives to the above tasks, there is no real alternative to item 1 in terms of conventional software you will use. Items ranked 2-3 all have alternatives, e.g. Python, Matlab, C++, etc.

• Unfortunately, as far as I'm aware, there is no good competitor to Mathematica (education license $1k; student license $70/year). Maple is probably the closest, but it was previously unsatisfactory for my needs.
• There is no good open source alternative…as far as I know. Perhaps we should do a benchmark test?