Alright, this is pretty cool. You can’t normally watch YouTube on a PS Vita (due to Sony’s browser not running Flash and YT not being set up for the console) but this is a neat workaround.
Basically, open the Vita browser, and go to this page. Then click and hold on “PSVitaTubeDownloader” to ‘open’ the link. Nothing will happen, that’s fine.
Finally, the good bit – go to YouTube as normal in the Vita browser, search for a video and then try to view it. It won’t work, but then click on the Vita browser ‘heart’ icon for the bookmarks and you’ll get a download link above where the video should be.
Then, via the browser, download the video to your Vita. It’ll be in the Videos app.