What does Mathematica do that Wolfram Alpha can't? It isn't clear to me what Mathematica does that Wolfram Alpha can't do.  (Obviously I have never used Mathematica.)  
Here (http://www.wolframalpha.com/faqs4.html) it says that Wolfram Alpha "has access to all of Mathematica's algorithms, which cover algebra, calculus, geometry, number theory, discrete math, and much more."  But clearly there must be more to Mathematica, or nobody would pay for it.
So what else can Mathematica do?
 A: You can give it much more precise commands, in a nutshell. You can't store variables in W|A, which, as a programmer, makes things harder. You can't get it to do certain things, like make graphs (like vertex-edge, not x-y plot). If you want a particularly complicated function plotted, W|A often refuses to do it. And perhaps most importantly, it's missing the "Manipulate" feature, which is fantastic for presenting a lot of things. (I was at a lecture about the unsolvability of quintics, and the lecturer wrote a program to show how the roots of $x^5 - x + t$ move as he varied $t$, and it was just beautiful).
A: Wolfram Alpha cannot do a lot of things that Mathematica can do.
This is true even though WA uses Mathematica as its backend.
At first, I thought you could enter any Mathematica command into WA and it would work. It is the furthest thing from the truth.
You can go to the Mathematica documentation, pick a built-in function and give it a go.
Some work however, and I think they are adding more functionality.
My guess is that they are trying to serve two different markets here.
There are other things too like programming, time for calculations, interfacing to other programs and just too much to list.
