First mover advantage. Youtube was synonymous with online 'canned' video, google with 'search'.
By parking it under the google brand they may have made a mistake. Another - possibly small - effect of that is that you have two steps before getting to a site, video.google.com, is less convenient than youtube.com, and much less easy to promote as a brand separate from the search portion of the site. Many people read 'video.google.com' the same way they promoted 'images.google.com', as a search engine for online videos (which it now has become, for the most part) instead of an easy way to share your videos with your buddies and the rest of the world.
I don't think it will be possible to quantify this effect, but I notice that microsoft named their search engine 'bing.com' after several tries of doing it as a subsidiary, and that google has not attempted to bring youtube.com under the google domain as the replacement for video.google.com (which still exists), it is now the 'search' arm of google for video, returning youtube.com mixed with other video results, just like what you'd expect.