Wolfram Alpha competes more with Wikipedia, Metaweb's Freebase, True Knowledge, and any natural language search engines (such as Microsoft's upcoming search engine, based perhaps in part on Powerset's technology among others), and other services that are trying to build comprehensive factual knowledge bases.
Wolfram Alpha is also different from Semantic Web systems for filtering the Web, organize knowledge, and track their interests.
The Mathematica engine it is currently built into Walfram Alpha, so the math part of it is rock solid.
Wolfram Alpha was able to solve novel numeric sequencing problems, calculus problems, and could answer questions about the human genome too. It was also able to compute answers to questions about many other kinds of topics (cooking, people, economics, etc.).
Wolfram Alpha is a system for computing the answers to questions. To accomplish this it uses built-in models of fields of knowledge, complete with data and algorithms, that represent real-world knowledge.
For example, it contains formal models of much of what we know about science -- massive amounts of data about various physical laws and properties, as well as data about the physical world.
Based on this you can ask it scientific questions and it can compute the answers for you. Even if it has not been programmed explicity to answer each question you might ask it.
But science is just one of the domains it knows about -- it also knows about technology, geography, weather, cooking, business, travel, people, music, and more.
It also has a natural language interface for asking it questions. This interface allows you to ask questions in plain language, or even in various forms of abbreviated notation, and then provides detailed answers.
The vision seems to be to create a system wich can do for formal knowledge (all the formally definable systems, heuristics, algorithms, rules, methods, theorems, and facts in the world) what search engines have done for informal knowledge (all the text and documents in various forms of media).
Wolfram's team manually entered, and in some cases automatically pulled in, masses of raw factual data about various fields of knowledge, plus models and algorithms for doing computations with the data. By building all of this in a modular fashion on top of the Mathematica engine, they have built a system that is able to actually do computations over vast data sets representing real-world knowledge. More importantly, it enables anyone to easily construct their own computations -- simply by asking questions.
The scientific and philosophical underpinnings of Wolfram Alpha are similar to those of the cellular automata systems he describes in his book, "A New Kind of Science" (NKS). Just as with cellular automata (such as the famous "Game of Life" algorithm that many have seen on screensavers), a set of simple rules and data can be used to generate surprisingly diverse, even lifelike patterns. One of the observations of NKS is that incredibly rich, even unpredictable patterns, can be generated from tiny sets of simple rules and data, when they are applied to their own output over and over again.
Speculating on Future Convergence
Could the Wolfram Alpha system actually be implemented using a cellular automata system ?
Quantum dot cellular automata using single atom room temperature quantum dots would seem likely to be blazingly fast.
Zyvex has a project to make lots of precise quantum dots.
Other opinion and first release date
I think there is more potential to this system than Stephen has revealed so far. I think he has bigger ambitions for it in the long-term future. I believe it has the potential to be THE online service for computing factual answers. THE system for factual knowlege on the Web. More than that, it may eventually have the potential to learn and even to make new discoveries. We'll have to wait and see where Wolfram takes it.
Maybe Wolfram Alpha could even do a better job of retrieving documents than Google, for certain kinds of questions -- by first understanding what you really want, then computing the answer, and then giving you links to documents that related to the answer. But even if it is never applied to document retrieval, I think it has the potential to play a leading role in all our daily lives -- it could function like a kind of expert assistant, with all the facts and computational power in the world at our fingertips.
I would expect that Wolfram Alpha will open up various API's in the future and then we'll begin to see some interesting new, intelligent, applications begin to emerge based on its underlying capabilities and what it knows already.
In May, Wolfram plans to open up what I believe will be a first version of Wolfram Alpha.
the Wolfram Alpha site
Stephen Wolfram blogging about Wolfram Alpha
One Ars technica take on Wolfram Alpha