Genie...Treewind's point...and mine... is that IF you want to keep a video, whether one YOU originally uploaded, or just one you'd like to keep, you have to use one of several tactics....BECAUSE...it is played from a temporary file in your browser cache, and will be deleted when the browser is closed (and sometimes as soon as you move away from that video and to another page).
   This means you must either have a separate program (such AS Firefox download helper or several othere) which will sort out the coded address and put the YT video where you want it OR you must go to where that temporary storage file is and move it before you close your browser.

I have used a couple of the download programs, but since video sites often rewrite the code to try to defeat them, I have learned where the cache is in my various browsers and how to identify the video files. As the videos are about finished, I 'pause' them and use a 'file manager' (like Explorer, only better) program to locate them by date, time & size, rename them, and move them to a place that is safe from automatic deletion.

I'm sure YT has their reasons for wanting us to view stuff ONLY while on their pages, but we have our reasons for wanting control over how & when they are played...(for example, being able to page thru a video one frame at a time and be able to capture a single frame.)