So, as happy as I am with the new YouTube app for the Vita, I'm happy that I have an alternative, because I find it irritating that I have to buffer the videos every single time. I have crappy internet, so it bugs me.
PS Vita YouTube Downloader!
What I can do is download the videos to my memory card from YouTube in the web browser. This method is not mine, but I would like to share it with you guys because it helped a lot of people. This method was also featured in IGN and a bunch of videos. I'll try to make it simple.
Type this url into your web browser:
tap and hold PS Vita Tube Downloader, and wait for... nothing.
Now go to YT. Choose a video you want to download it. It will still say that you do not have an html5 browser and it isn't supported, but just open the bookmark that you created a while ago. One or two small links will appear directly above the video, saying 480 or 720. Click on these and the download will start onto your memory card. Enjoy :)
Note: Some videos don't have HD so a 720 pixels option may not be available.