Symptoms

1. When you try to run a video in mp4 format from a browser (usually Safari) inside an iPad or Iphone it doesn't work. (Web Issue).
2. Sometimes when you try to run an mp4 format video inside my iPad/iPhone (not on the web) it doesn't work as well. (Local Issue)

You can read about this issue on the official discussion site from apple here.

Diagnose

One important thing to note is that "iDevices" doesn't support all the varieties of mp4. Here is a small list about MPEG-4 and H.264 supported varieties.

According to Apple official website, iPad or iPhone 4S is only compatible with H.264 or MPEG-4 video format with the following specifications:

• If it is H.264 video, it should meet: up to 1080p, 30 frames per second, High Profile level 4.1 with AAC-LC audio up to 160 Kbps, 48kHz, stereo audio in .m4v, .mp4, and .mov file formats;
• If it is MPEG-4 video, it should meet: up to 2.5 Mbps, 640 by 480 pixels, 30 frames per second, Simple Profile with AAC-LC audio up to 160 Kbps per channel, 48kHz, stereo audio in .m4v, .mp4, and .mov file formats;

But there are some forums saying that in fact, the iPad specifications are these:

iPad Supported Video Format

• H.264 video (up to 720p, 30 frames per second; main profile level 3.1 with AAC-LC audio up to 160 Kbps, 48kHz, stereo audio in .m4v, .mp4, and .mov file formats)
• MPEG-4 video, up to 2.5 Mbps, 640 by 480 pixels, 30 frames per second, Simple Profile with AAC-LC audio up to 160 Kbps, 48kHz, stereo audio in .m4v, .mp4, and .mov file formats.

 Please also note these HLS requirements by apple. Notably interlaced video is not supported.

Cause

1. (Web Issue) - happens because the iPad/iPhone doesn't have an HTML5 player that works so well as it should yet.
2. (Local Issue) - happens for a completely different cause, which is the supported resolution of the mp4 videos on "iDevices".

Workaround

1. (Web Issue) - Download the video to your device. The "iDevices" are quite unstable with mp4 format and h.264 on HTML5 (this is not yet solved by apple, sometimes the videos work, sometimes they don't. This is related to the "varieties" we were talking above). This is going to solve the problem unless you end up walking into the second Issue.
2. (Local Issue) - Your mp4 video resolution is not supported by the "iDevice", so you will have to reduce the video resolution to 720p or convert the video (you can convert on HandBrake). Here is a very complete blog post about how to encode the video and what is supported by the phones. Here is another article explaining how to watch and convert your mp4 videos.