Wolfram Alpha is a computational engine. That means they don't just crawl the web and index the crap out of it to make it easily searchable, like Google does. They collect and organize that data so you can perform complex calculations. WA also has the functions of graphing and scientific calculators.
Historically, they went after different demographics. Wolfram Alpha has been treated more like a calculator, for graphing, solving, and cross-referencing complex sets of data. Google's been used for... well, searching. They coexisted peacefully. But things got competitive when Apple used Wolfram Alpha to power Siri. Suddenly, if you wanted to quickly look up a fun fact on your iPhone, you no longer went to Google. You just asked Siri, and Wolfram Alpha gave you an answer.
Google had 3 things to say about that.
One, they launched their own graphing calculator last winter which allowed people not only to visualize mathematical formulas but to zoom and pan around the 3D graphs.
Two, Google launched the Knowledge Graph. This is the biggest direct competitor to Wolfram Alpha. Instead of just offering the 10 blue links, Google can now answer many of your questions directly. Google will continue to expand the Knowledge Graph, but for now, it is still limited compared to WA. For example, both Google and Wolfram Alpha can tell you how old Sergey Brin is (38), but only WA can tell you how old Sergey Brin was when Google was founded (25!).
Three, Google has just added a full 34 button scientific calculator to its search results page. And the coolest (nerdiest) part is, you can calculate by voice. Even on your desktop computer, just click the microphone button and speak away. As soon as you type in an equation, the calculator will pop up above the regular search results, giving you access to almost all functions you'd find on an expensive graphing calculator.
But perhaps an even cooler part is that the full functionality of this calculator is available on your smartphone. As soon as you type in your math problem, the calculator will pop out along with the answer. Keep the calculator in portrait mode (on the left) and you'll only see the numbers and basic functions like add, subtract, divide, and multiply. But turn your phone sideways (on the right), and you get the full assortment of advanced tools like tangent and square root.
The calculator can also be brought up just by typing "calculator" into the search box on your phone or computer.