Hello Katsu,
there is really a (browser) problem: By trying to insert your mp4 file into a sketch I found a strange phenomenon.
Please have a look here again, I modified it: http://www.hansgd.de/AdobTest/_qu/indexTab.html.
The QuickTime video activated by the link shows the complete control panel. The other one, embedded in a table for test purposes shows only the running button. Changing height or width has not improved the result.
That means there is a real (browser) problem: When parsing HTML, the browser assumes and obeys some default capabilities (properties like title, poster, controls, muted, autoplay, loop, etc.) as prescribed by the HTML way of displaying content within a page. In its bare essence, it displays the file by just using the browser and the system’s playback features (a simplified QuickTime view). And, by the way, there's no difference in IE or FF!
For this reason, I would suggest: please convert the mp4 file to another format. I prefer flv or swf files. Both you can provide with a control panel.
What concerns the source code: Please use "right click" and you will see, that I only used used your file and added some command lines.
Good luck!
Hans-Günter