Apache out of the box
Apache2 does not support streaming out of the box however nowadays most browsers do, that means if you have correctly setup your MIME applications browsers will decide what course of action to take based on MIME type. That means things like Mp4 will work using a compatible codec installed on the users machine, often things like VLC can be installed on users machines supporting even more compatibility. You should encounter problems when the runtime of the video is long and attempting to skip far forward results in having to wait a long time depending on what broadband connection your using, as it will download the whole file while proper streaming only downloads what it needs.
Mod h246 benefits
mod_h246 module has a few benefits such as faster skipping to parts (Timeshifting seek) and can be made even more compatible by using Adobe Flash which then bypasses the requirements for users to have the appropriate codecs as Adobe will be decoding it for you. It also helps save on bandwidth, and allows you to create playlists type files that play a section of a video rather than the whole lot... for example if you had a 1 hour video but you only want users to be able to view 10mins of it, then this is possible.
YouTube, Vimeo and so on...
One of the main reasons so many websites use YouTube, Vimeo and so on is because of the compatibility to work seamlessly on all browsers and devices, it'll even trans-code the video in different formats meaning even people on mobile devices will have sensible bitrate, as desktop bitrate is much higher. Unless there's a major reason why you can't host on these networks I'd advise to do so and mark the videos private.