Often the reason that a video cannot be played on TED.com is because it came from a "TEDx" event. You can learn more about TEDx here:
https://www.ted.com/participate/organize-a-local-tedx-event
Occasionally, some exemplary TEDx talks are chosen by our content team to be featured on TED.com as a Talk of the Day, like this one:
https://www.ted.com/talks/shawn_achor_the_happy_secret_to_better_work
When a TEDx talk is featured as a Talk of the Day, that elevates it to "official TED talk" status. The speaker is given an official bio by our content team, and the talk is added as a file on TED.com (as opposed to an embed). This makes it downloadable and viewable everywhere.
At the same time, we're also in the process of moving all TED-related content from YouTube onto TED.com, so that it can be part of a larger, holistic TED library. This will eventually include all TEDx and TED-Ed videos. However, moving this content over does not entail featuring it as a Talk of the Day, so "official TED talk" status is not granted to any content added to the library in this process.
This non-featured content is only embedded on TED.com as YouTube videos, not added as unique files. This means that the content isn't downloadable, it won't have subtitles, and any restrictions upon YouTube content will also apply. In other words, if a school network blocks YouTube, or a local ISP does not allow access to YouTube, the content will not be viewable even though it is shown on TED.com.
This non-featured TEDx and TED-Ed content is also not available on the TED Android or iOS app.