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?