How many different versions do exist of this movie?
There are different versions of "The Day After" available. Amongst other things, the reason for that were differences between studio and director, restrictions from the network plus an originally different design for the movie with a running time of more than 4 hrs. It was supposed to be a two-parter but it finally became a movie with a running time of approx. 2 hrs. followed by a discussion. For that matter 50% of the footage needed to be removed / was scratched from production schedule. If and when the 4 hour version (or at least the unknown footage, e.g. as deleted scenes on a future release) is going to released is unknown. Not even the 25th anniversairy in November 2008 was a reason for MGM or ABC to come up with a new release which is weird because those kinds of anniversairies are usually the perfect opportunity for a re-release of a classic.There are five different versions availabe and two of them were officially released in the Western world: the first DVD Edition aka "Modern Version" and the DVD re-release which equals the first edition on VHS. A detailed comparison between the "Modern Version" and the first edition VHS release, split in two parts, can be found here (part 1) and here (part 2).
Is this any relation to the recent blockbuster, 'The Day After Tomorrow?'
No. "The Day After" attempts to show the very plausible aftermath, should a nuclear war occur. The discussion after the original broadcast also introduced us to the concept of 'nuclear winter'. (see trivia) "The Day After Tomorrow" is essentially a disaster movie, which speculates global warming as the cause of catastrophic worldwide weather disruptions. "The Day After" attempts to present a "scientifically accurate" cause and effect, while "The Day After Tomorrow" relies on scientific evidence as a basis for highly unlikely events.