The end date has very much truth and relevance, but it means nothing about the "end of society", or any post-apocalyptic Earth situation.
The long count ends every 26,000 years, not 13,000, thats only the halfway mark.
This cycle refers to the precession of the Earth - the tilt on our axis, pointing the planet in different directions of the galaxy.
There's no myth here.. more like Math. Check out
this page to see it broken down a little more. Its not simple stuff here.
There will be no visible changes in human society such as asteroids and shit smashing up the planet. but the mayans were never wrong in astronomical ideas when it came to mathematics based problems - NEVER.. and most scholars think a global shift in consciousness will occur on the most basic of levels. We're already seeing it now as the planet moves away from a technological revolution and more into a techno-spiritual one where technology has become a part of us and is no longer foreign.
This is science, not religion. I would read up on the end date if you're interested.. I don't know how trustable that History Channel special was by your post.