While studying at Princeton, I chose Maple over Mathematica for several reasons:
(1) My father bought me a copy of Maple for my laptop --- having Maple right there made a huge difference --- I originally learned about Maple from using Mathcad.
(2) Maple (in the early '90s) had better official and unofficial documentation than Mathematica.
(3) The Maple language is similar to Fortran --- actually quite similar to modern Fortran 2003/2008 --- I would like to see Maple build a better bridge between Maple and Fortran --- i.e. code generation for Fortran 2003/2008 and a Compile command for compiling a Maple procedure with Fortran. Intrinsically, there is a close connection between the evolution of Fortran and Maple in the past 15 years even if not intentional.
(4) The open source aspect of Maple including infolevel made Maple much more useful --- plus it was easy to see many examples of advanced Maple programming. Maple could do more to make others understand the excitement of Maple programming drawing upon its language and its data types. Maple should consider updating and adopting A. Heck's book as the standard Maple book --- it was one of the very best --- much better than what Mathematica ever had. Also, First Leaves was such a fun read --- the current documentation lacks the excitement of the early days of computer algebra and the computer algebra wars.
(5) I beta tested Maple in 1994 for Maple V R4, and hence, I helped select the features of the Classic Worksheet. Currently, I still use the Classic Interface because it has editting features that are missing in Standard. I would like to see the Classic Interface added as an option in the Standard Interface. It could be marketed like Coke Classic --- that was a big hit :).
(6) Maple has typically covered a broader range of mathematics than Mathematica with an eye towards conforming to to traditional mathematical notation.
(7) Shift-Enter: One of the reasons (strange as it may sound) that I chose Maple over Mathematica was that you could use Enter rather than Shift-Enter to enter expressions and run programs. In the Standard Interface Maple adopted the Mathematica convention. Why I ask? Maple had it right the first time. Maple should re-exmine and treamline the keystrokes required to perform computations in the Standard Worksheet. They should be competitive with the Classic Interface or even better --- and certainly they should be better than Mathematica. Maple V R2 and R4 had simple interfaces like modern Mathematica and modern web browsers like Chrome. Just as browsers have simplied their buttons, Maple should move to a more modern streamlined interface --- at least such an updated Classic interface should be an option within the modern Java Standard Interface. Making an interface for advanced Maple would (I think) show even entry users the potential power of Maple in comparison with showy but less substantial alternatives.
(8) Site License: The site license at Princeton also helped me learn Maple because I could run some hefty jobs on the server. I wish Maple would press harder to put Maple in every major University. I would like to see a site license at the University of Chicago where I am a professor. Some of my students select Mathematica because there is a site license --- I try to encourage them to try Maple.
Depertment of ChemistryUniversity of Chicago