Going back to the original communication model I outlined earlier, the critical difference between the two definitions becomes clear. Free speech is defined in terms of the endpoints, in terms of the rights of the senders and receivers. Censorship is defined in terms of control over the medium.
The methods of suppressing free speech and the methods of censoring are very different. Suppression of free speech tends to occur through political or legal means. Someone is thrown in jail for criticizing the government, and the police exert their power to remove the controversial content from the Internet. On the receiver's side, consider China, which is an entire country who's government has decided that there are publicly available sites on the Internet that will simply not be available to anybody in that country, such as the Wall Street Journal. Suppressing free speech does not really require a high level of technology, just a high level of vigilance, which all law enforcement requires anyhow.
Censorship, on the other hand, is taking primarily technological forms. Since messages flow on the Internet at speeds vastly surpassing any human's capabilities to understand or process, technology is being developed that attempts to censor Internet content, with generally atrocious results. (A site called Peacefire has been good at documenting the failures of some of the most popular censorware, as censoring software is known.) Nevertheless, the appeal of such technology to some people is such that in all likelihood, money will continue to be thrown at the problem until some vaguely reasonable method of censorship is found.