caniuse.com is a great resource for pretty good data to answer this question.
At the moment ... ~ 92% of web browsers support the HTML video tag . The main one, which is not Opera Mini (about 4.5%). For these users, you can use a Flash backup, which actually doesn't work too much. There are some very simple solutions that will handle this for you, such as videoJS, jPlayer and JWPlayer.
Now you need to encode two, maybe three formats. About 92% of users support MPEG-4 / h.264 . Opera Mini and IE8 do not support it.
Only about 71% of users can support full-screen video , so for Android and iOS (basically) all versions. Best of all, you can set the video to fill 100% of the browser window. If full screen mode is very important, then you will want to use Flash.
So, in short, yes, HTML5 video requires a bit of extra work, but at the moment it is not so difficult, and it is a standard that is moving towards better stability and uniformity. For example, YouTube uses it (with backups) if it indicates that it is ready in prime time.
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