Which browser are you using? Are you on 4.1 (Jelly Bean) or ICS?

 

The default browser should see Flash content in ICS. All of them should recognize HTML5.

If possible and the video isn't private, could you share a link to a site that doesn't work?

 

Makes me wonder if the iPhone is just downloading a stream of the video, since it can't save video, it does the only thing it can. Since the Nexus can save the video and the video might not be flagged as stream-able, it assumes its just any old file and downloads it. It might have nothing to do with flash or HTML5.

Just a guess though since I don't know what the site is.

 

Here's my YouTube tutorial. Hopefully it helps.

[TUTORIAL] How to install Adobe Flash on the Google Nexus 7 Tablet - YouTube

 

Correct me if im wrong but i dont believe the iphone has EVER had flash capabilities it plays html format but not flash. Android also plays html format however if you are trying to view flash content its best that you stick with the default browser and make sure you have the "plugins" option set to enable.

The reason that i suggest that you stick to the default browser is because at least one of the browsers that you mentioned above (chrome) is incapable of playing flash content. If you are still unable to view the video set your browser to desktop mode or manually enter the "www" desktop web address for the web site.

There is no reason that i can think of as to why your video wont play???

 

Actually did you download "Adobe Air" or "Adobe Flash"? that could be the root of your problem. By the way if you do have adobe flash make sure not to delete it because Adobe has taken it out of the play store for jellybean devices. They actually plan on taking it off the store period and honestly im not sure if they already have or not which is why i'm asking if you have air or flash.

 

Tried to download the exact same file from the exact same guy in your vid, and the file has been removed.

Edit: Same file is over on XDA (see file attachment with QR code) - http://forum.xda-developers.com/showpost.php?p=23663853&postcount=1286

 

Dolphin has an iPhone user agent, you could see if that allows you to play the video.