Both talks were TEDx, which means they are independently organized TED events. According to Emily McManus, editor of TED, these talks are not vetted, but they can be removed from the TEDx archive if they contain ideas that are "wrong to the point of being unscientific."
According to their science board, Rupert Sheldrake's argument was based on what they refer to as "several major factual errors." Sheldrake was not given the opportunity to rebut this assertion before his presentation was taken down.
Regarding Graham Hancock's talk, TED claimed he misrepresents modern scientific thought with the suggestion that "no scientists are working on the problem of consciousness." However, nowhere in his talk does Hancock make this statement. TED also deemed his observations about psychotropic drugs to be "both nonscientific and reckless," though they offered no explaination for their opinion. They also claimed that Hancock called psychotropic drug use essential for "an emergence into consciousness." A review of the presentation makes clear that no such assertion is ever made.
TED's removal of the videos from YouTube has provoked wide condemnation. As journalist and author Robert McLuhan writes in his blog, "that's what TED is all about: to present challenging new ideas and innovative thinking. If it doesn't take risks and rile the establishment sometimes, what's it for?"
On the TED blog, Kent Bye offers a comprehensive post on the history of the controversy, dating back to December, 2012, when materialists in the TED community began to protest the presence of presenters who they considered overly spiritual at TEDx events.