Thanks Jack for the suggestion, but I think I will stick to Firefox 22, rather than revert back to an earlier version. The reason is that I also use IE9 which does show the Real Downloader button. However, when I download music videos from YouTube they still do not play. So, I do not feel it is worthwhile going back to Firefox 22 just to get the Real Downloader button working as I do not believe the videos will play.
I have tried playing music videos downloaded with Real Downloader using IE9 in Windows Media Player but they do not play in that format either. I have not installed Media Player Classic. I might look into that, but I fear that they would not play in that player either.