There are two different ideas behind what you see in Maple and in Mathematica. Generally speaking (and there are exceptions here and there), in Maple there is an intentional emphasis in giving you control as much as possible over every step of a computation, while not going 'static'. Among two good examples of that, 1) in Maple you then have inert representations for mostly everything, you can compute with integrals, derivatives, or functions without ever computing the integrals, derivatives of the functions themselves, just taking into account their properties; 2) in Maple, products of powers of the same base - say a^n * a^m - are not automatically combined. In some sense this is equivalent to requesting from you to sometimes guide the system more specifically in this or that direction. On the other hand, in Mathematica, preference is given to perform these computations automatically, assuming that you are not interested in controlling any of these steps.
So, for example, if in Mathematica you enter Exp[x/2]*Exp[-3/2*x] the exponentials are automatically combined. while if you try the same in Maple, exp(x/2)*exp(-3/2*x), these two factors are not combined. But then if you add exp(x/2), in Maple you can factor this sum easily, as in
> factor(exp(x/2) + exp(x/2)*exp(-3/2*x)));
while in Mathematica you get the less convenient result
In:= Factor[Exp[x/2] + Exp[-3*x/2]*Exp[x/2]]
Out= E^-x (1 + E^(x/2)) (1 - E^(x/2) + E^x)
So sometimes it is convenient to combine products of powers of the same base, sometimes not.
The corollary: yes these systems are different in a number of things, there are advantages and disadvantages in both approaches, and depending on the example one approach is more convenient than the other one. Taking that into account, in each system the commands you need to enter to obtain a result are not the same. I cannot claim neutrality as I work for Maplesoft but, anyway, in my opinion I really like the control maple offers over the computations, and I also like options to switch gears and compute in a more automatic mode. In the context of the Physics package, for instance, one new option (so far experimental) lets you 'automatically combine powers of the same base', another will optionally let you compute derivatives with respect to complex variables using Wirtinger operators instead of the standard classical complex analysis, and etc.
Edgardo S. Cheb-Terrab