For all the mp4 files in here,
JWPlayer reproduces the audio track but the video is black. By using Apple QuickTime instead it plays.
The mp4 here
instead works well with JWPlayer.
The files were generated from different systems. Maybe some encoding parameter is different - how can I find out what's wrong?

The videos in the first link use an unsupported video codec (MPEG4 Part 2 aka MPEG4 Visual aka ISO/IEC 14496-2) while the second set use a supported video codec. Adobe Flash only supports H.264 aka Advanced Video Codec aka MPEG4 Part 10 when using the MP4/F4V container.
In the future you can find this info using tools like e.g. Media Info.