How to start and record Flash video stream without recording the player taskbar
As long as the video isn't copy protected, you will want to use a stream ripper instead of trying to do a screen capture of your monitor.  Any number of software products are able to directly transfer a video that is in the Flash streaming video format as long as it isn't copy protected.  A quick search for "flash video ripper" and your operating system should return numerous results to choose from.  The advantage of this is that the stream can be recorded with the original encoding preserved, which avoids loss of quality due to repeat encoding and compression.

As a side bonus, since the stream rippers will record the stream directly, you also don't have to worry about the bar or your mouse being over the screen.