I have some jQuery tabs one of which holds a flash video. When I play the video in one tab and click to another in FF or Safari the video stops along with the sound, clicki
It's likely because you're applying the display: none;
to the div holding the flash object, not the flash object itself.
I was messing around with a sort of similar issue today and while I don't have an exact solution for you I do have a possibility you could try when you jump between divs to try and turn the player off:
$(flashobjectid).attr('height', '0');
If the flash player you're using happens to be the commonly used jw player then you could do (when you hide the div):
document.getElementById(playerId).sendEvent("PLAY","false");