I had a similar issue and came across this post after I figured out my issue.
CyanBlue's suggestion would have put me on the right track.
I was able to see the skin of my FLVPlayback component when testing on my local HD, but when I published to the web it wasn't there. The problem was that I didn't understand the nature of the skin SWFs... in fact I didn't even notice that when I generated the main SWF that a secondary SWF was also generated [for the skin] & it needed to be uploaded as well. Once I uploaded the additional SWF, then the FLVPlayback had what it needed. As Cyan mentioned, you'll need to have this in the same directory as the primary SWF.
I worked around this by using the individual components (i.e. Play, Mute, Vol, etc...) which was a good excercise... but those can be a little quirky it seems.
In the process I also found a good Adobe reference to customizing the skin components: http://www.adobe.com/devnet/flash/ar...lvcomp_04.html
There you can get into the sample FLA files and see how they work and modify as desired.
Controversial as the site may be... here's the link
in case anyone cares to see something that they may have never heard about.