If you don't use the same "video id number" in the old and new URLs, then your site will quickly become difficult to maintain, and will get slower and slower every time you add a new video URL, because each new video will require an additional rewriterule.
Your example URLs above aren't consistent enough for me to diagnose the problem with your code, but note this important point: Mod_rewrite cannot change the links on your pages. All it can do is to detect requests for certain URLs, and pass those requests to your script with the URL-path information appended as name/value pairs.
The closest I can get is to take you example URLs and filepath above, but assume that the "video id number" does not change: # Internally rewrite requests for "/video?x=y" to filepath "/watch_video.php?x=y" RewriteRule ^video$ /watch_video.php [L]
Note that with this code, the "v=2238HDRSN3NY" query string is passed through the rule unchanged. As coded, this query string may contain arbitrary name/value pairs or it may be blank.