I'm in the camp that considers a demo almost essential before buying a game. In addition to all the things mentioned that pertain to the "gameplay" aspects (technical, as "how is the control scheme (keyboard customization, etc.)" to artistic, as "how interesting is the story", "how are the animations, sounds, etc."), there is also the "compatibility" aspect. Will the program run on my computer? If yes, how well does it run?
None of these compatibility aspects can be gauged by watching a trailer. Anyone replying "read the required/recommended specs for the game" should take a look at all the possible hardware configurations (graphic, sound, cpu, memory) and compare this to the simple list that is usually presented as the game's specs. Those specs are a rough guidance, but nothing more. How often do you have forum posts that go "My computer is way beyond recommended specs, I can play similar titles on highest settings, but game X just crawls on my PC."
I can't play Gothic 4 (Arcania) on my PC, because there seems to be a problem with my graphic card (confirmed by forum posts from users with the same problem). Amnesia runs at around 0.05 fps. Pathologic repeatedly crashed (on my previous PC) after 5 minutes with some sound error. To be precise, I don't know if the above is true, I should have said that the *demos* of those games behaved that way. Nothing of this I would have ever known if I had only watched trailers of those games.
Why seem demos to have come out of flavor? The video game industry has matured and expanded its target population since 10-15 years before. Budgets have risen immensely. You simply can't reach that many people by a demo as you can with "traditional" advertising online, in game mags or on tv. Advertising/PR is always about *control* of information flow. From big budget productions like Starcraft II and now Diablo we get carefully timed information snippets.
Ten years ago, a publisher website might have a 10 day countdown to the release of a new game. Today, they have 10 day countdown to the *announcement* of a new game. 15 years ago, discussion was about doctored screenshots in preview material. A typical explanation was that graphical feature X was planned at that time, not yet implemented, so faked on the preview screenshot. Later feature X was canceled, but nobody seemed to have informed the PR department. So a faked screenshot ended up not only in the preview material (understandable) but on the back of the box. 10 years ago we had back of the boxes that consisted of 6 pictures, 5 of them from cut scences and one from actual gamplay. Today we have the word "gameplay trailer", which only exists because there are other trailers that do not contain gameplay elements, something that would not have made sense to me 10 years ago.
Game marketing is now in the same state as marketing of other consumer goods: It misleads anywhere it can, tries to sell a product not by its features (that would be boring) but by some emotional grip. With the advent of the internet and widespread availability of costumer reviews for almost anything, it is now much more important that you put the message "Buy this game!" into the heads of as many people as possible at launch. A pre-release game demo would directly undermine that, because the potential customer could now form his own opinion. A post-release game demo is probably seen as a waste from a marketing perspective, as all the buzz is already over. This ties in greatly with the fact that games work like hollywood movies: The "opening weekend" is all important and after a few month sales have probably trickled down to almost nothing.