SymPy is a Python library for symbolic computation. It provides computer algebra capabilities either as a standalone application, as a library to other applications, or live on the web as SymPy Live or SymPy Gamma.
License: Open Source
Mathematica is a powerful commercial software with extensive capabilities in symbolic and numerical computation, advanced algorithms, and a user-friendly interface, but it comes with licensing costs. SymPy, on the other hand, is an open-source Python library that is more flexible for integration with other Python applications, making it suitable for educational and research purposes, though it may lack some of the advanced features found in Mathematica.