I remember that back when I was in University there was a set of simultaneous ordinary differential equations that MAPLE was able to solve that Mathematica choked on.

I remember reporting this to my tutor. He looked at me with a steely gaze and said “That is a very serious allegation to make, young man. Are you willing stand by it?”

Maple's PDE solving abilities always made me want to yell "witchcraft!".

P.S. I actually remembered one area where Maple is still absolutely superior, that is plotting. It doesn't require to explicitly set argument range, it tries to find features of the function and adjusts the range to include them. AFAIK, Mathematica still doesn't do this.