I agree with this distinction, and free speech and democracy are certainly not the same. Democracy in its most basal form is majority rule, e.g. distribution of power via some sort of election system. In a sense this is more than a "form of government", since it entails the rights to elect and to be elected.
However, political scientists generally agree that for democracy to have meaning there is a cluster of other rights that are needed to make sure that people can actually elect someone based on their preferences or interests, and these rights include freedom of speech, of organization/congregation, rule of law, some form of minority protection, etc. Of course, these rights are generally not seen as absolute, and no democratic society that I know allows you to falsely accuse someone in public of being a child molester or yell "fire" in a cinema without fear of persecution.
Moreover, history has shown us that free speech is most under threat in the less democratic countries. The first thing a modern dictator does is grab control over the "old" centralized media, and controlling new media is a logical next step for the more tech-savvy dictators. If you compare the "freedom of the press" and "freedom house" reports, it is hard not to see the correlation (e.g. http://imgs.ntd.tv/content/201... vs http://rsf.org/index2014/data/...)
In other words: yes democracy and freedom of speech are separate concepts, but they are very strongly related.
(also, the importance to democracy of platforms such as twitter is *vastly* overblown, but that is a different discussion :))
(note that geeks often have difficulty understanding non-absolute rights. There is a big difference between free speech in Russia and in Britain, even if Britain curbs free speech with strong libel laws. Interpreting and upholding the non-absolute rights requires strong institutions including independent judiciary, responsive politics, and critical media. No country has a fully independent judiciary, fully responsive politics, or fully critical media, but for the love of god go visit Russia (or Algeria, or Uzbekistan, or China) and tell me how much you appreciate the US/European judiciary, politics, and media. "Our" institutions go out of line sometimes, but there are strong mechanisms for keeping them in check if they wander too far from what is deemed acceptable)