Keane from YouTube announced in a YouTube Help thread that they decided to remove the tags from showing up on the video watch page because some are abusing it.
Keane said it wasn't used much by the average user and when used, it was typically abused by being copied from video to video. He wrote:
Tags no longer appear on this page - this isn't a bug, but a change that went out this week. Having them on the watch page, in some cases, gave users an opportunity to abuse tags by copying them from other videos. We also didn't see much usage of tags by the average viewer.
In no way does this mean you should not tag your videos. YouTube will continue to use it for discovery and search purposes. Keane explained:
Though tags don't appear on the video page, anyone who uploads should keep adding them to your videos. Tags are still used to aid the discoverability of your videos like they have before, the only difference now is that they don't appear on the video page.
Forum discussion at YouTube Help.