CASes generate symbolic derivative expressions. This is typically a fairly expensive operation, and can be slow if the number of expressions are large or if the expressions are complicated. You have to pre-generate these expressions before you can do any work (paying an upfront cost). CAS'es also use a very general symbolic engine, which entails a performance penalty.

If you're doing numerical work and don't need explicit derivative expressions, automatic differentiation offers a method that is both fast and convenient.